1. Cornerstones Series Editors Charles L. Epstein, University
of Pennsylvania, Philadelphia Steven G. Krantz, Washington
University, St. Louis Advisory Board Anthony W. Knapp, State
University of New York at Stony Brook, Emeritus
2. J.A.C. Kolk Distributions Theory and Applications Translated
from Dutch by J.P. van Braam Houckgeest J.J. Duistermaat
3. J.J. Duistermaat Mathematical Institute Utrecht University
J.A.C. Kolk Utrecht University P.O. Box 80.010 3508 TA Utrecht
Mathematical Institute The Netherlands [email protected] Springer
New York Dordrecht Heidelberg London ISBN 978-0-8176-4672-1 e-ISBN
978-0-8176-4675-2 DOI 10.1007/978-0-8176-4675-2 Mathematics Subject
Classification (2010): 46-01, 42-01, 35-01, 28-01, 34-01, 26-01
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4. To V. S. Varadarajan A True Friend and Source of
Inspiration
7. Preface I am sure that something must be found. There must
exist a notion of generalized functions which are to functions what
the real numbers are to the rationals (G. Peano, 1912) Not that
much effort is needed, for it is such a smooth and simple theory
(F. Tr`eves, 1975) In undergraduate physics a lecturer will be
tempted to say on certain occasions: Let .x/ be a function on the
line that equals 0 away from 0 and is innite at 0 in such a way
that its total integral is 1. The most important property of .x/ is
exemplied by the identity Z 1 1 .x/.x/ dx D .0/; whereis any
continuous function of x. Such a function .x/ is an object that one
frequently would like to use, but of course there is no such
function, because a function that is 0 everywhere except at one
point has integral 0. All the same, it is important to realize what
our lecturer is trying to accomplish: to describe an object in
terms of the way it behaves when integrated against a function. It
is for such purposes that the theory of distributions, or
generalized functions, was created. It can be formulated in all
dimensions, its mathematical scope is vast, and it has
revolutionized modern analysis. One way to elaborate on the
distributional point of view1 is to note that a point- wise
denition of functions is not very relevant to many situations
arising in engi- neering or physics. This is due to the fact that
physical observations often do not represent sharp computations at
a single point in space-time but rather averages of uctuations in
small but nite regions in space-time. This is an essential point in
signal theory, where there are limitations to the determination of
pulse lengths, and 1 Here we follow the masterly exposition of
Varadarajan [22, p. 185]. ix
8. x Preface in quantum theory, where the electromagnetic elds
of elementary particles cannot be measured unless one uses a
macroscopic test body. From the mathematical point of view one can
say that a measurement of a physical quantity by means of a test
body yields an average of the values of that quantity in a very
small region, the lat- ter being represented by a smooth function
that is zero outside a small domain. One replaces the test bodies
by these functions, which are naturally called test functions. The
value thus measured is a function on the space of test functions,
and the inter- pretation of the measurement as an average makes it
clear that this function must be linear. Thus, if T is the space of
test functions (unspecied at this point), phys- ical quantities
assign real or complex values to functions in T . In keeping with
our idea that measurements are averages, we recognize that
sometimes things are not so bad and that actual point measurements
are possible. Thus ordinary functions are also allowed to be viewed
as functionals on T . If f is such an ordinary function, it
represents the following functional on T :7! Z f .x/.x/ dx 2 C:
However, since we admit measurements that are too singular to be
represented by ordinary functions, we refer to the general
functionals on T as generalized func- tions or distributions. We
have been vague about what the space is in which we are operating
and also what functions are chosen as test functions. This actually
is a great strength of these ideas, because the methods evidently
apply without any re- striction on the nature or the dimensions of
the space. In this book, however, we restrict ourselves to the most
important case, that of open subsets of the Euclidean spaces Rn .
Distributions are to functions what the real numbers R are to the
rational numbers Q. In R, the cube root of any number also belongs
to R, as does the logarithm of the absolute value of a nonzero
number; by contrast, 3 p 2 and log 2 do not belong to Q. Moreover,
R is the smallest extension of Q having such properties, while
every real number can be approximated by rationals with arbitrary
precision. Similarly, distri- butions are always innitely
differentiable, which is not true of all functions. Here, too,
distributions are the smallest possible extension of the test
functions satisfying this property, while every distribution can be
approximated in the appropriate sense by test functions with
arbitrary precision. Continuing the analogy, we mention that
differential equations may have distributional solutions in
situations where there are no classical solutions, that is, given
by differentiable functions. In numerous prob- lems it is of great
advantage that solutions exist, even at the penalty of introducing
new objects such as distributions, because the solutions can be
subject to further study. The theory of distributions provides many
tools for the investigation of these so-called weak solutions; for
example, these tools enable one to determine when and where
distributions are actually functions. One of the early triumphs of
distribution theory was the result that every partial differential
equation with constant coef- cients has a fundamental solution in
the sense of distributions: classically, nothing comparable is
available.
9. Preface xi Fourier theory is another branch of analysis in
which a suitable subclass of all distributions helps to clarify
many issues. This theory is a far-reaching generaliza- tion of
writing a vector x D .x1; : : : ; xn/ in Rn as x D nX kD1 xk ek;
that is, as a superposition of a nite sum of multiples of the basis
vectors ek. Anal- ogously, in Fourier analysis one attempts to
write functions or even distributions as superpositions of basic
functions. In this case, nitely many functions do not sufce, but
the collection of all bounded exponential functions turns out to be
a good choice: bounded, because unbounded exponentials grow too
fast at innity, and exponential, because such functions are
simultaneous eigenvectors of all partial derivatives. The sense in
which the innite superposition represents the original object then
becomes an important issue: is the convergence pointwise or
uniform, or in a smeared sense? Fourier analysis in the
distributional setting enables one to han- dle problems that
classically were out of reach, as well as many new ones. So one
obtains, working modulo 2, .x/ D 1 2 1X kD 1 eikx : This formula
goes back to Euler, except that he found the sum to be equal to 0
when x is away from 0. Hormanders monumental treatise [11] on
linear partial differential equations and Harish-Chandras
pioneering work [10] on harmonic analysis on semisimple Lie groups
over the elds of real, complex, or p-adic numbers are but two of
the rich fruits borne by Schwartzs text [20], which gave birth to
the theory of distributions. This book aims to be a thorough, yet
concise and application-oriented, intro- duction to the theory of
distributions that can be covered in one semester. These
constraints forced us to make choices: we try to be rigorous but do
not construct a complete theory that prepares the reader for all
aspects and applications of distribu- tions. It supplies a certain
degree of rigor for a kind of calculation that people long ago did
completely heuristically, and it establishes what is legitimate and
what is not. The amount of functional analysis that is needed in
our treatment is reduced to a bare minimum: only the principle of
uniform boundedness is used, while the HahnBanach theorems are
applied to give alternative proofs, with one exception, of results
obtained by different methods. On the other hand, in our exposition
of the theory and, in particular, in the problems, we stress
applications and interactions with other parts of mathematics. As a
result of this approach our text is complementary to the books [13]
and [14] by A.W. Knapp, also published in the Cornerstones series.
Building on rm foundations in functional analysis and measure
theory, Knapp develops the theory rigorously and in greater depth
and wider context than we do, by treating pseudodif- ferential
operators on manifolds, for instance. In many ways our text is
introductory;
10. xii Preface on the other hand, it presents students of
(theoretical) physics or electrical engineer- ing with an idea of
what distributions are all about from the mathematical point of
view, while giving applied or pure mathematicians a taste of the
power of distribu- tions as a natural method in analysis. Our aim
is to make the reader familiar with the essentials of the theory in
an efcient and fairly rigorous way, while emphasizing the
applications. Solutions of important ordinary and partial
differential equations, such as the equation for an electrical LRC
network, those of CauchyRiemann, Laplace, and Helmholtz and the
heat and wave equations, are studied in great detail. Tools for the
investigation of the regularity of the solution, that is, its
smoothness, are developed. Topics in signal reconstruction have
also been treated, such as the mathematical theory underlying CT (=
computed tomography)scanners as well as results on band- limited
functions. The fundamentals of the theory of complex-analytic
functions in one variable are efciently derived in the context of
distributions. In order to make the book self-contained, various
results on special functions that are used in our treatment are
deduced as consequences of the theory itself, wherever possible. A
large number of problems is included; they are found at the end of
each chap- ter. Some of these illustrate the theory itself, while
others explore its relevance to other parts of mathematics. They
vary from straightforward applications of the the- ory to theorems
or projects examining a topic in some depth. In particular,
important aspects of multidimensional real analysis are studied
from the point of view of dis- tributions. Complete solutions to
146 of the 281 problems are provided; problems for which solutions
are available are marked by the symbol. A great number of the
remaining exercises are supplied with copious hints, and many of
the more difcult problems have been tested in take-home
examinations. In more technical terms, the rst eleven chapters
cover the basics of general dis- tributions. Specically, Chap. 10
presents a systematic calculus of pullback and pushforward for the
transformation of distributions under a change of variables,
whereas Chap. 13 considers complex-analytic one-parameter families
of distribu- tions with the aim of obtaining fundamental solutions
of certain partial differen- tial operators. Chap. 14 then goes on
to treat the Fourier transform of the subclass of tempered
distributions in the general, aperiodic case, which is of
fundamental importance for the subsequent Chaps. 1519. Chap. 15
discusses the notion of a distribution kernel of a continuous
linear mapping. This notion enables an elegant verication of many
properties of such mappings. More generally, it enables aspects of
the theory of distributions to be surveyed from a fresh and
unifying point of view, as is exemplied by many of the problems in
the chapter. The Fourier inversion formula is used in a novel proof
of the Kernel Theorem. The Fourier transform is applied in Chap. 16
to study the periodic case and in Chap. 17 to construct addi-
tional fundamental solutions. Chap. 18 deals with the Fourier
transforms of com- pactly supported distributions, and Chap. 19
considers rudiments of the theory of Sobolev spaces. Mathematically
sophisticated readers, having perused the rst ten chapters, might
prefer to proceed immediately to Chaps. 14 and 15.
11. Preface xiii Important characteristics of the present
treatment of the theory of distributions are the following. The
theory as presented provides a highly coherent context with a
strong potential for unication of seemingly distant parts of
analysis. A systematic use of the operations of pullback and
pushforward enables the development of a very clean and concise
notation. A survey of distribution theory in the framework of dis-
tribution kernels allows a description that is algebraic rather
than analytic in nature, and makes it possible to study
distributions with a minimal use of test functions. In particular,
within this framework some more advanced aspects of distribution
theory can be developed in a highly efcient manner and transparent
proofs can be given. The treatment emphasizes the role of symmetry
in obtaining short arguments. In ad- dition, distributions
invariant under the actions of various groups of transformations
are investigated. Our preferred theory of integration is that of
Riemann, because it will be more familiar to most readers than that
of Lebesgue. In some instances, however, our arguments might be
slightly shortened by the use of Lebesgues theory. In the very
limited number of cases in which Lebesgue integration is essential,
we mention this explicitly. The reader who is not familiar with
measure theory may safely skip these passages. On the other hand,
in the theory of distributions Radon measures arise naturally as
linear forms dened on compactly supported continuous functions, and
therefore the Daniell approach to the theory of integration, which
emphasizes linear forms acting on functions instead of functions
acting on sets, is very natural in this set- ting. In the Appendix,
Chap. 20, we survey the theory of Lebesgue integration with respect
to a measure from this point of view. Although the approach seems
very appropriate in our context, we are aware of the fact that it
is of limited value to the mathematical probabilist, who primarily
requires a theory of integration on function spaces, which are not
usually locally compact. We strongly feel that a mathematical style
of writing is appropriate for our pur- poses, so the book contains
a certain amount of theoremproof text. The reader of a text at this
level of mathematical sophistication rightly expects to nd all the
information needed to follow the argument as well as clear
expositions of dif- cult points, and the theoremproof format is a
time-honored vehicle for conveying these. Furthermore, in theorems
one summarizes useful information for future ap- plication.
Important results (for instance, the Fourier inversion formula)
often get several proofs; in this manner different aspects or
unexpected relations are brought to the fore. The present text has
evolved from a set of notes for courses taught at Utrecht Uni-
versity over the last twenty years, mainly to bachelor-degree
students in their third year of theoretical physics and/or
mathematics. In those courses, familiarity with measure theory,
functional analysis, or even some of the more theoretical aspects
of real analysis, such as compactness, could not be assumed. Since
this book addresses the same type of audience, the present text was
therefore designed to be essentially self-contained: the reader is
assumed to have merely a working knowledge of linear algebra and of
multidimensional real analysis (see [7], for instance), while only
a
12. xiv Preface few of the problems also require some
acquaintance with the residue calculus from complex analysis in one
variable. In some cases, the notion of a group will be en-
countered, mainly in the form of a (one-parameter) group of
transformations acting on Rn . Each time the course was taught, the
notes were corrected and rened, with the help of the students; we
are grateful to them for their remarks. In particular, J.J. Kuit
made a considerable number of original contributions and we
benetted from fruit- ful discussions with him. M.A. de Reus
suggested many improvements. Also, we express our gratitude to our
colleagues E.P. van den Ban, for making available the notes for his
course in 1987 on distributions and Fourier transform and for very
constructive criticism of a preliminary draft, and R.W. Bruggeman,
for the improve- ments and additional problems that he contributed
over the past few years. In ad- dition, T.H. Koornwinder read
substantial parts of the manuscript with great care when preparing
a course on distributions, and contributed signicantly, by many
valuable queries and comments, to the accuracy of the nal version.
Furthermore, we wish to acknowledge our indebtedness to A.W. Knapp,
who played an essential role in the publication of this book, for
his generous advice and encouragement. The enthusiasm and wisdom of
Ann Kostant, our editor at Birkhauser, made it all possible, and we
are very grateful to her for this. Jessica Belanger saw the
manuscript through its nal stages of production. The original Dutch
text has been translated with meticulous care by J.P. van Braam
Houckgeest. In addition, his comments have led to considerable
improvement in formulation. The second author is very thankful to
M.J. Suttorp and H.W.M. Plokker, cardiol- ogists, and their teams:
their intervention was essential for the completion of this book.
The responsibility for any imprecisions remains entirely ours; we
would be grate- ful to be told of them, at [email protected].
Utrecht, Hans Duistermaat February 2010 Johan Kolk
13. Standard Notation The symbol set against the right margin
signies the end of a proof. Furthermore, the symbol marks the end
of a denition, example or remark. Item Meaning ; empty set o, O
little and big O symbol of Landau i p 1 x 2 X or X 3 x x an element
of X x X x not an element of X f x 2 X j P g the set of x in X such
that P holds @X, X boundary and closure of the set X XY or YX X a
subset of Y X [ Y , XY , X n Y union, intersection, difference of
sets XY Cartesian product of sets f W X ! Y , x 7! f .x/ mapping,
effect of mapping f .; y/ mapping x 7! f .x; y/ f .X/, f 1 .X/
direct and inverse image under f of the set X f jX restriction to X
g f composition of f and g, or of g following f Z, Q, R, C
integers, rationals, reals, complex numbers Za integers greater
than or equal to a N D Z1, natural numbers Ra reals larger than a
jxj absolute value of x 2 R x greatest integerx sgn x sign of x a;
b open interval from a to b a; b closed interval from a to b a; b ,
a; b half-open intervals .a; b/ column vector in R2 .x1; : : : ;
xn/ column vector xv
14. xvi Standard Notation kxk norm of vector x h x; y i inner
product of vectors x and y Rn , Cn spaces of column vectors Re z,
Im z real and imaginary parts of complex z z complex conjugate of z
jzj absolute value of z 2 C f .a/ function f W Rn ! C given by x 7!
f .ax/ f 0 , f 00 derivative and second derivative of f W R ! R f
.k/ kth-order derivative of f Df (total) derivative of mapping f W
Rn ! Rp @j f partial derivative of f with respect to jth variable#
0approaches 0 through positive valuesP , Q sum and product,
possibly with a limit operation I identity matrix or operator det A
determinant of matrix or operator A t A transpose of matrix or
operator A ' is isomorphic to, is equivalent to
15. Chapter 1 Motivation Distributions form a class of objects
that contains the continuous functions as a sub- set. Conversely,
every distribution can be approximated by innitely differentiable
functions, and for that reason one also uses the term generalized
functions instead of distributions. Even so, not every distribution
is a function. In several respects, the calculus of distributions
can be developed more read- ily than the theory of continuous
functions. For example, every distribution has a derivative, which
itself is also a distribution (see Chap. 4). Hence, every
continuous function considered as a distribution has derivatives of
all orders. Conversely, we shall prove that every distribution can
locally be written as a linear combination of derivatives of some
continuous function (see Theorem 13.1 or Example18.2). If ev- ery
continuous function is to be innitely differentiable as a
distribution, no proper subset of the space of distributions can
therefore be adequate. In this sense, the ex- tension of the
concept of functions to that of distributions is as economical as
it possibly can be. This may be compared with the extension of the
system Z of integers to the system Q of rational numbers, where to
any x and y 2 Z with y 0 corresponds the quotient x y 2 Q. In this
case, too, Q is the smallest extension of Z having the desired
properties. We now discuss some more concrete types of problem and
show how they are solved by the calculus of distributions. We also
indicate some typical contexts in which these questions arise. It
should be pointed out that the reader will not be assumed to be
familiar with the nonmathematical concepts used in those contexts.
Likewise, in Examples 1.1 through 1.5 we will occasionally use some
mathematical tools that are not yet assumed to be known by the
reader. The point of these examples is to provide insight into
where the subject is going, rather than to give all details about
each example. Example 1.1. Here we consider the second-order
derivative of a function that is non- differentiable at one point.
The function f dened by f .x/ D jxj for x 2 R is continuous on R.
It is differentiable on R0 and on R0, with derivative equaling 1
and C1 on these 1 Springer Science+Business Media, LLC 2010
Cornerstones, DOI 10.1007/978-0-8176-4675-2_1, J.J. Duistermaat and
J.A.C. Kolk, Distributions: Theory and Applications,
16. 2 1 Motivation intervals, respectively. f is not
differentiable at 0. So it seems natural to say that the derivative
f 0 .x/ equals the sign sgn.x/ of x, the value of f 0 .0/ not being
dened and, intuitively speaking, being of little importance. But
beware, we obviously require that the second-order derivative f 00
.x/ equal 0 for x0 and for x0, while f 00 .x/ must have an
essential contribution at x D 0. Indeed, if f 00 .x/0, the
conclusion is that f 0 .x/c for a constant c 2 R; in other words, f
.x/ D c x for all x 2 R, which is different from the function x 7!
jxj, whatever choice is made for c. The correct description turns
out to be that f 00 D 2, with as in the preface; see Problem 4.1
for more details. Example 1.2. Now we are concerned with the
electrical eld of a point charge. In Maxwells theory of
electromagnetism there are physical difculties with the con- cept
of a point charge, and in its mathematical description a problem
occurs as well. Let v W R3 ! R3 be a continuously differentiable
mapping, interpreted as a vector eld on R3 . Further, let x 7!
divv.x/ D 3X jD1 @vj .x/ @xj be the divergence of v; this is a
continuous real-valued function on R3 . Suppose that X is a bounded
and open set in R3 having a smooth boundary @X and lying at one
side of @X; we write the outer normal to @X at the point y 2 @X as
.y/. The Divergence Theorem then asserts that Z X div v.x/ dx D Z
@X hv.y/; .y/i dyI (1.1) see for instance DuistermaatKolk [7,
Theorem 7.8.5]. Here the right-hand side is interpreted as an
amount of volume that ows outward across the boundary, while div v
is rather like a local expansion (= source strength) in a motion
whose velocity eld equals v. Traditionally, one also wishes to
allow point (mass) sources at a point p, for which R X div v.x/ dx
D c if p 2 X and R X div v.x/ dx D 0 if p X, where X is the closure
of X in R3 . (We make no statement for the case that p 2 @X.) Here
c is a positive constant, the strength of the point source in p.
These conditions cannot be realized by a function divv continuous
everywhere on R3 . More specically, the divergence of the special
vector eld v.x/ D 1 kxk3 x (1.2) vanishes at every point x 0;
verify this. This implies that the left-hand side of (1.1) equals 0
if 0 X and 4 if 0 2 X; the latter result is obtained when we
replace the set X by X n B, where B is a closed ball around 0,
having a radius sufciently small that BX, and then compute the
right-hand side of (1.1) (see [7, Example 7.9.4]). Thus we would
like to conclude that div v in this case equals the point source at
the point 0 with strength 4; in mathematical terms, div v D 4
,
17. 1 Motivation 3 with the generalization to R3 of the from
the preface (see Problem 4.6 and its solution for the details).
Example 1.3. The underlying mathematical background of this example
is the the- ory of the Hilbert transform H, which gives a way of
describing a negative phase shift of signals by 90 (see Problem
14.52), that is, .H cos/.x/ D sin x D cosx2; .H sin/.x/ D cos x D
sinx2: In addition, in the example we come across interesting new
distributions that play a role in quantum eld theory. The function
x 7! 1 x is not absolutely integrable on any bounded interval
around 0, so it is not immediately clear what Z R .x/ x dx is to
mean ifis a continuous function that vanishes outside a bounded
interval. Even if .0/ D 0, the integrand is not necessarily
absolutely integrable. For example, let 0c1 and dene (see Fig. 1.1)
.x/ D 0 if x0; 1 j log xj if 0xc: This functionis continuous on 1;
c and can be extended to a continuous 0.5 1.4 0.5 50 Fig. 1.1
Graphs ofand x 7! .x/ x function on R that vanishes outside a
bounded interval. Then Z c.x/ x dx D log j log j log j log cj; and
the right-hand side converges to 1 as# 0. But ifis continuously
differentiable and vanishes outside a bounded interval, integration
by parts and the estimate ./ . / D O./ as# 0, which is a
18. 4 1 Motivation consequence of the Mean Value Theorem, give
lim #0 Z Rn ; .x/ x dx D Z R 0 .x/ log jxj dx: (1.3) Note the
importance of the excluded intervals being symmetric about the
origin. The left-hand side is called the principal value of the
integral of x 7! .x/ x and is also written as PV Z R .x/ x dx DWPV
1 x./: (1.4) More generally, if c 2 R andis a continuous function
on R, denePV 1 x c./ D lim #0 Z Rn c ; cC .x/ x c dx; provided that
this limit exists. Other, equally natural, propositions can also be
made. Indeed, always assumingto be continuously differentiable and
to vanish outside a bounded interval, Z R .x/ x C idx converges as#
0, or 0, respectively; see Problem 1.3. The limit is denoted by Z R
.x/ x C i 0 dx; (1.5) or Z R .x/ x i 0 dx; (1.6) respectively.
Clearly, the functions PV 1 x , 1 xCi 0 , and 1 x i 0 differ only
at x D 0, that is, integration against a functionwill produce
identical results if .0/ D 0. In Problem 1.3 and its solution as
well as in Examples 3.3 and 5.7 one may nd more information.
Example 1.4. We will now make plausible that the theory of
distributions provides limits in cases where these do not exist
classically and that it also enables more freedom in the
interchange of analytic operations. For 0r1 and x 2 R, summation of
the geometric series leads toP n2Z0 .rei x /n D 1 1 rei x . By
taking the real parts in this identity we obtain X n2Z0 rn cos nx D
1 r cos x 1 C r2 2r cos x DW Ar.x/: Our interest is in the behavior
of the preceding identity when r D 1 or under tak- ing limits for
r1. First consider the series for r D 1. Then it is clearly
divergent, to 1, if x 2 2Z. In fact, the series is divergent
everywhere on R. This follows from
19. 1 Motivation 5 the fact that for no x 2 R do we have cos nx
! 0 as n ! 1. Indeed, otherwise we would have sin2 nx D 1 cos2 nx !
1, but also sin2 nx D 1 2 .1 cos 2nx/ ! 1 2 . On the other hand,
lim r1 Ar .x/ D 1 2 .x 2 R n 2Z/ and Ar .x/ D 1 1 r .x 2 2Z/: Abels
Theorem (see [13, Theorem 1.48]), which would imply P n2Z0 cos nx D
1 2 for x 2 Rn2Z, does not apply, because of the divergence
everywhere of the series. 0.1 0.05 0.05 0.1 10 20 2 2 10 20 Fig.
1.2 Graph of Ar , for r D 9 Pk jD1 10 j with 2k5, and of A0:99999
Nevertheless the numerical evidence in Fig. 1.2 above strongly
suggests that the sum of the series is given by the function having
value 1 2 on R n 2Z and 1 on 2Z. However, in the theory of
integration as discussed in Chap. 20 such a function would be
identied with the constant function 1 2 on R, which ignores the
serious divergence of the series on 2Z. Therefore, it might be more
reasonable to describe the limit of the series as A1 WD 1 2 C c P
k2Z 2k on R. Here c 2 C is a suitable constant and 2k denotes the
Dirac function located at 2k. We determine c by demanding lim r1 Z
Ar .x/ dx D Z A1.x/ dx: For 0r1, an antiderivative Ir of Ar is
given by (see Fig. 1.3 below) Ir .x/ D x 2 C arctan 1 C r 1 r tan x
2I so Z Ar .x/ dx D X Ir ./ D 2; which is also directly obvious by
termwise integration of the series. On the other hand, R A1.x/ dx
DC c, and so c D ; phrased differently, X n2Z0 cos nD 1 2 CX k2Z 2k
on R: The validity of this equality in the sense of distributions
is rigorously veried in Problem 16.8.
20. 6 1 Motivation 2 2 Fig. 1.3 Graph of Ir , for r D k 10 with
0k10, and of an antiderivative of 1 2 Note that I1.x/ WD limr1
Ir.x/ D x 2 , for x 0. Accordingly R A1.x/ dx D P I1./. In
addition, A1 is the derivative in the sense of distributions of I1
(classically the latter is nondifferentiable at 0), as will be
shown in Example 4.2. According to the theory of integration we
would have the limit of functions limr1 Ar D 1 2 (the union of the
lower solid line segment and the dashed line seg- ment in Fig. 1.3
is the graph of an antiderivative of 1 2 ) and then lim r1 Z Ar.x/
dx D 2 D Z lim r1 Ar .x/ dx: (1.7) In view of the Dominated
Convergence Theorem, Theorem 20.26.(iv), of Lebesgue or that of
Arzel`a (see [7, Theorem 6.12.3]), the fact that interchange of
limit and integration in (1.7) is invalid means that the family
.Ar/0r1 does not admit an integrable majorant on ;; see Fig. 1.4.
In other words, there exists no function g on R that is integrable
on ;, while jAr .x/jg.x/, for all 0r1 and x 2 ;. This is
corroborated by the fact that near 0 the envelope, see [7, Exercise
5.38], of the graphs of the Ar, for 0r1, is given by the graph of x
7! 1 2 .1 C 1 j sin xj /; this function is not integrable on
bounded intervals containing 0. Furthermore, with similar arguments
as above, one derives the equality of distri- butions on R X n2Z
einx D 2 X k2Z 2k.x/: This is the correct form of Eulers formula
from the preface and gives in essence the basic result in the
theory of Fourier series; see (16.7). In addition, we have the
following equality of distributions on a sufciently small open
neighborhood of 0, where we use the principal value from Example
1.3:
21. 1 Motivation 7 0.75 1 2 Fig. 1.4 Graph of Ar , for r D k 10
with 0k9 X n2N sin nx D x cos x 2 2 sin x 2PV 1 xD 1 2 log.1 cos x/
0 : Note that the function in front of PV has limit 1 as x ! 0 and
that log.1 cos/ is integrable near 0, while x 7! 1 x is not.
Further, limn!1 sin nx D 0 if and only if x 2 Z. Indeed, under the
assumption we obtain 2 sin x cos nx D sin.n C 1/x sin.n 1/x ! 0,
that is, sin x D 0 by the preceding result about cos. Finally, we
observe that the family of functions .Ar / is closely related to
the Poisson kernel .Pr / (see (16.12)), which plays an important
role in boundary value problems for the Laplace equation. Example
1.5. Another motivation, signicant also for historical reasons, has
its root in the calculus of variations, the theory of nding optimal
solutions. An idea shared by most craftsmen, artists, engineers,
and scientists is the principle of economy of means.
Mathematically, this is the principle of least action and the
theory of the as- sociated differential equations of EulerLagrange.
These variational equations form the basis for many mathematical
models in the sciences and in economics. Solv- ing them can be a
daunting task; in addition, in the nineteenth century doubts arose
about the existence of solutions in the general case. One might
think that nature does not pose articial problems and that the
applied mathematician therefore need not worry about these matters.
A physical theory, however, is not a description of na- ture but a
model of nature that may well be troubled by mathematical
difculties. For instance, the description of the electric eld near
a very sharp charged needle poses problems both mathematically and
physically: the actual experiment produces sparking. The starting
point for the rather lengthy discussion is an obvious calculation.
This is then followed by an existence theorem concerning minima of
functions, the full proof of which is not allowed by the present
context, however. The discussion ends in the statement of a problem
that will be solved by means of distribution theory at a later
stage. Consider F.v/ D 1 2 Z b a p.x/ v0 .x/2 C q.x/ v.x/2dx;
22. 8 1 Motivation where p and q are given nonnegative and
sufciently differentiable functions on the interval a; b . Let C k
; be the set of k times continuously differentiable functions v on
a; b with v.a/ D and v.b/ D . For k1, we consider F to be a
real-valued function on C k ; . We now ask whether among these v a
special u can be found for which F reaches its minimum, that is, u
2 C k ; and F.u/F.v/ for all v 2 C k ; . If such u is obtained, one
nds that for every2 Ck 0; 0, the function t 7! F.u C t/ attains its
minimum at t D 0. This implies that its derivative with respect to
t at t D 0 equals 0, or Z b a p.x/ u0 .x/ 0 .x/ C q.x/ u.x/ .x/dx D
0: If u 2 C2 , integration by parts gives Z b ad dx .p.x/ u0 .x// C
q.x/ u.x/.x/ dx D 0: Since this must hold for all2 Ck 0; 0, we
conclude that u must satisfy the second- order differential
equation .Lu/.x/ WD d dx .p.x/ u0 / C q.x/ u D 0: (1.8) This
procedure may be applied to much more general functionals
(functions on spaces of functions) F ; the differential equations
that one obtains for the stationary point u of F are called the
EulerLagrange equations. Until the middle of the nineteenth century
the existence of a minimizing u 2 C2 was taken for granted.
Weierstrass then brought up the seemingly innocuous example a D 1,
b D 1, D 1, D 1, p.x/ D x2 , and q.x/ D 0. One may then consider,
for any 0, the function v.x/ D arctan.x=/ arctan.1=/ ; for x 2 1; 1
. The denominator has been included in order to guarantee that v
.1/ D 1. For x0, or x0, we have that arctan.x=/ converges to =2, or
C=2, respectively, as# 0. Therefore, v converges to the sign
function sgn as# 0. To study the behavior of F .v/ we write v0 .x/
D 1arctan.1=/ 1 1 C .x=/2 : The change of variables x Dy leads to F
.v/ D1 arctan2.1=/ Z 1= 1= y2 2.1 C y2/2 dy:
23. 1 Motivation 9 Note that in this expression the factor 1
arctan2.1=/ converges to 4=2 as# 0. One has 2y2 .1 C y2/2 D 0 .y/
if .y/ D y 1 C y2 C arctan y: That makes the integral equal to
..1=/ . 1=//=4, from which we can see that the integral converges
to =4 when# 0. The conclusion is that F .v/ D./, where ./ converges
to 1= as# 0. In particular, F .v/ converges to zero as# 0. Thus we
see that the inmum of F on C1 1; 1 equals zero; indeed, even the
in- mum on the subspace C 1 1; 1 equals zero. However, if u is a C1
function with F.u/ D 0, we have du dx .x/0, which means that u is
constant. But then u cannot satisfy the boundary conditions u. 1/ D
1 and u.1/ D 1. In other words, the restriction of F to the space C
1 1; 1 does not attain its minimum in this example. In the
beginning of the twentieth century the following discovery was
made. Let H.1/ be the space of the square-integrable functions v on
a; b whose derivatives v0 are also square-integrable on a; b .
Actually, this is not so easy to dene. A cor- rect denition is
given in Chap. 19: v 2 H.1/ if and only if v is square-integrable
and the distribution v0 is also square-integrable. In order to
understand this denition, we have to know how a square-integrable
function can be interpreted as a distribution. Next, we use that
the derivative of any distribution is another distribution, which
may or may not equal a square-integrable function. If v is a
continuously differentiable function on a; b , application of the
CauchySchwarz inequality (see [7, Exercise 6.72]) gives jv.x/ v.y/j
D Z x y v0 .z/ dz Z x y v0 .z/2 dz 1=2 Z x y dz 1=2kv0 kL2 jx yj1=2
; (1.9) where kv0 kL2 is the L2 norm of v0 . This can be used to
prove that every v 2 H.1/ can be interpreted as a continuous
function on a; b (also compare Example 19.3), with the same
estimate jv.x/ v.y/jkv0 kL2 jx yj1=2 : The continuity of the
functions v 2 H.1/ implies that one can meaningfully speak of the
subspace H; .1/ of the v 2 H.1/ for which v.a/ D and v.b/ D . Also,
for every v 2 H.1/ the number F.v/ is well-dened. Now assume that
p.x/0 for every x 2 a; b ; this excludes the example of
Weierstrass. The assumption implies the existence of a constant c
with the property kv0 k2 L2c F.v/; for all v 2 H.1/. In combination
with the estimate for jv.x/ v.y/j this tells us that every sequence
.vj /j2N in H; .1/ with bounded values F.vj / is an equicon-
24. 10 1 Motivation tinuous and uniformly bounded sequence of
continuous functions. By the Arzel`a Ascoli Theorem (see Knapp [13,
Theorem 10.48]), a subsequence .vj.k//k2N then converges uniformly
to a continuous function u as k ! 1. A second fact here of- fered
without proof is that u 2 H ; .1/ and that the values F.vj.k//
converge to F.u/ as k ! 1. This is now applied to a sequence of vj
for which F.vj / converges to the inmum i of F on H ; .1/ . Thus
one can show the existence of a u 2 H ; .1/ with F.u/ D i. In other
words, F attains its minimum on H; .1/ . This looked promising, but
one then ran into the problem that initially all one could say
about this minimizing u was that u0 is square-integrable. This does
not even imply that u is differentiable under the classical
denition that the limit of the difference quotients exists. Because
so far we do not even know that u 2 C 2 , the integration by parts
is problematic and, as a consequence, so is the conclusion that u
is a solution of the EulerLagrange equation. What we can do is to
integrate by parts with the roles of u andinterchanged and thereby
conclude that Z b a u.x/ .L/.x/ dx D 0; (1.10) for every2 C 1 that
vanishes identically in a neighborhood of the boundary points a and
b. For this statement to be meaningful, u need only be a locally
in- tegrable function on the interval I D a; b . In that case the
function u is said to satisfy the differential equation Lu D 0 in a
distributional sense. Historically, a somewhat older term is in a
weak sense, but this is not very specic. Assume that p and q are
sufciently differentiable and that p has no zeros in the interval
I. In this text we will show by means of distribution theory that
if u is a locally integrable function and satises the equation Lu D
0 in the distributional sense, u is in fact innitely differentiable
in I and satises the equation Lu D 0 on I in the usual sense. See
Theorem 9.4. In this way, distribution theory makes a contribution
to the calculus of variations: by application of the Arzel`aAscoli
Theorem it demonstrates the existence of a minimizing function u 2
H; .1/ . Every minimizing function u 2 H ; .1/ satises the
differential equation Lu D 0 in the distributional sense;
distribution theory yields the result that u is in fact innitely
differentiable and satises the differential equation Lu D 0 in the
classical sense. This application may be extended to a very broad
class of variational problems, also including functions of several
variables, for which the EulerLagrange variation equation then
becomes a partial differential equation. Some of the interesting
phenomena in the preceding examples form our starting point for the
development of the theory of distributions. The estimation result
from the next lemma, Lemma 1.6, will play an important role in what
follows. The functions in Examples 1.1, 1.2, and 1.3 are not
continuous, or differentiable, respectively, at a special point.
Singularities in functions can be mitigated by trans- lating the
function f back and forth and averaging the functions thus obtained
with
25. 1 Motivation 11 a weight function .y/ that depends on the
translation y applied to the original func- tion. Let us assume
thatis sufciently differentiable on R, that .x/0 for all x 2 R,
that a constant m0 exists such that .x/ D 0 if jxjm, and nally,
that Z R .x/ dx D 1: (1.11) For the existence of such , see Problem
1.4. The averaging procedure is described by the formula .f/.x/ D Z
R f .x y/ .y/ dy D Z R f .z/ .x z/ dz: (1.12) The minus sign is
used to obtain symmetric formulas; in particular, f D f . The
function fis called the convolution of f and , because one of the
functions is reected and translated, then multiplied by the other
one, following which the result is integrated. Another
interpretation is that of a measuring device recording a signal f
around the position x, where .y/ represents the sensitivity of the
device at displacement y. In practice, thisis never completely
concentrated at y D 0; because of built-in inertia, .y/ will have
one or more bounded derivatives. Yet another interpretation is
obtained by dening Tyf , the function translated by y, via .Tyf
/.x/ WD f .x y/: (1.13) Here we use the rule that .Tyf /.x C y/,
the value of the translated function at the translated point,
equals f .x/, the value of the function at the original point. In
other words, under Ty the graph translates to the right if y0. If
we now read the rst equality in (1.12) as an identity between
functions of x, we have f D Z R .y/ Tyf dy: (1.14) Here the
right-hand side is dened as the limit of Riemann sums in the space
of continuous functions (of x), where the limit is taken with
respect to the supremum norm. Thus, the functions f translated by y
are superposed, with application of a weight function .y/, similar
to a photograph that becomes softer (blurred) if the camera is
moved during the exposure. Indeed, differentiation with respect to
x under the integral sign in the right-hand side in (1.12) yields,
even in the case that f is merely continuous, that f is
differentiable, with derivative .f/0 D f0 : In obtaining this
result, we have not used the normalization (1.11). We can therefore
repeat this and conclude that f is equally often continuously
differentiable as . For more details, see the proof of Lemma 2.18
below. How closely does the smoothed signal f approximate the true
signal f ? If af .z/b for all z 2 x m; x C m , we conclude from
(1.12) and (1.11)
26. 12 1 Motivation that a.f/.x/b as well. This can be improved
upon if we can bring the positive number m closer to 0. To achieve
this, we replace the functionby(see Fig. 1.5), for an arbitrary
constant 0, with .x/ D 1 x : (1.15) Furthermore,is equally often
continuously differentiable as . Fig. 1.5 Graph ofas in (1.15)
withequal to 1 and 1/2, respectively Lemma 1.6. If f is continuous
on R, the function f converges uniformly to f on every bounded
interval a; b as# 0. And for every 0, the function f on R is
equally often continuously differentiable as . Proof. We have Z R
.y/ dy D Z Ry dyD Z R .z/ dz D 1; from which .f/.x/ f .x/ D Z R f
.x y/ f .x/.y/ dy D Zmm f .x y/ f .x/.y/ dy; where in the second
identity we have used .y/ D 0 if jyj m. This leads to the estimate
j.f/.x/ f .x/jZmm jf .x y/ f .x/j .y/ dysup jyj m jf .x y/ f .x/j;
where in the rst inequality we have applied .y/0 and in the second
inequality we have once again used the fact that the integral
ofequals 1. The continuity of f gives that for every 00 the
function f is uniformly continuous on the bounded interval a 0; b C
0 (if necessary, see [7, Theorem
27. 1 Problems 13 1.8.15] taken in conjunction with Theorem 2.2
below). This implies that for every 0 there exists a 00 with the
property that jf .x y/ f .x/j if x 2 a; b and jyj. From this we may
conclude j.f/.x/ f .x/j, if x 2 a; b and 0=m. Bochner [3] has
called the mapping f 7! f an approximate identity, and Weyl [24], a
mollier. Problems 1.1. For acb, the integral R b a 1 x c dx is
divergent. Prove PV Z b a 1 x c dx D log b c c a : 1.2. Calculate
PV R R .x/ x dx, for the following choices of : 1.x/ D x 1 C x2 and
2.x/ D 1 1 C x2 : Which of these two integrals converges absolutely
as an improper integral? 1.3.Determine the difference between (1.4)
and (1.5), and between (1.5) and (1.6). Each is a complex multiple
of .0/. See Example 14.30 and Problem 12.14 for different
approaches. 1.4.Determine a polynomial function p on R of degree
six for which p.a/ D 1 1 35 32 Fig. 1.6 Illustration for Problem
1.4 p0 .a/ D p00 .a/ D 0 for a D 1, while in addition, R 1 1 p.x/
dx D 1. Dene .x/ D p.x/ for jxj1 and .x/ D 0 for jxj1. Prove thatis
twice continu- ously differentiable. Sketch the graph of(see Fig.
1.6). 1.5.Let be twice continuously differentiable on R and let
equal 0 outside a bounded interval. Set f .x/ D jxj. Calculate the
second-order derivative of g D
28. 14 1 Motivation 0.01 0.01 200 0.0015 0.0015 1 1 0.003 0.003
0.003 Fig. 1.7 Illustration for Problem 1.5. Graphs of g00 D 2p,
forD 1=100, and of g0 and g, forD 1=1000 fby rst differentiating
under the integral sign, then splitting the integration at the
singular point of f , and nally eliminating the differentiations in
every sub- integral. Now take equal towithas in Problem 1.4 andas
in (1.15). Draw a sketch of g00 for small , and, by nding
antiderivatives, of g0 and g. Show the sketches of g, g0 , g00 next
to those of f , f 0 , and f 00 (?), respectively (see Fig. 1.7).
1.6.We consider integrable functions f and g on R that vanish
outside the interval 1; 1 . (i) Determine the interval outside
which the convolution fg certainly vanishes. (ii) Using simple
examples of your own choice for f and g, calculate fg, and sketch
the graphs of f , g, and fg. (iii) Try to choose f and g such that
f and g are not continuous while fg is. (iv) Try to choose f and g
such that fg is not continuous. Hint: let 1, f .x/ D g.x/ D x if
0x1, f .x/ D g.x/ D 0 if x0 or x1. Verify that f and g are
integrable. Prove the existence of a constant c0 such that .fg/.x/
D c x2C1 if 0x1. For what values of is fg discontinuous at the
point 0? 4 3 2 2 3 4 Fig. 1.8 Illustration for Problem 1.7. Graph
of arccos B cos
29. 1 Problems 15 1.7. Set I.x/ D x, for all jxj1, and let
triangle W R ! R denote the unit triangle function given by
triangle.x/ D 1 jxj, for jxj1 and triangle.x/ D 0, for jxj1. Prove
(see Fig. 1.8) in the notation of Denition 2.17 that cos B arccos D
I on 1; 1 ; arccos B cos DX k2Z T.2kC1/ triangle D X k2Z T2k .1 0;
1 0; / on R: Hint: show that arccos B cos is continuous on R, while
on R nZ one has .arccos B cos/0 D sin j sin j D X k2Z . 1/k Tk 10;
:
30. Chapter 2 Test Functions We will now introduce test
functions and do so by specializing the testing of f as in (1.12).
If we set x D 0 and replace .y/ by . y/, the result of testing f by
means of the weight functionbecomes equal to the integral inner
product hf; i D Z R f .x/ .x/ dx: (2.1) (For real-valued functions
this is in fact an inner product; for complex-valued func- tions
one uses the Hermitian inner product hf; i.) In Chap. 1 we went on
to vary , by translating and rescaling. The idea behind the
denition of distributions is that we consider (2.1) as a func- tion
of all possible test functions , in other words, we will be
considering the mapping test f W7! Z R f .x/ .x/ dx: Before we can
do so, we rst have to specify what functions will be allowed as
test functions. The rst requirement is that all these functions be
complex-valued. Denition 2.5 below, of test functions, refers to
compact sets. In this text we will be frequently encountering such
sets; therefore we begin by collecting some informa- tion on them.
Denition 2.1. An open cover of a set K in Rn is a collection U of
open sets in Rn such that their union contains K. That is, for
every x 2 K there exists a U 2 U with x 2 U . A subcover is a
subcollection E of U still covering K. In other words, EU and K is
contained in the union of the sets U with U 2 E. The set K is said
to be compact if every open cover of K has a nite subcover. This
concept is applicable in very general topological spaces. Next,
recall the concept of a subsequence of an innite sequence
.x.j//j2N. This is a sequence having terms of the form y.j/ D
x.i.j// where i.1/i.2/; in particular, limj!1 i.j/ D 1. Note that
if the sequence .x.j//j2N converges to x, every subsequence of this
sequence also converges to x. , Springer Science+Business Media,
LLC 2010 J.J. Duistermaat and J.A.C. Kolk, Distributions: Theory
and Applications, 17 Cornerstones, DOI
10.1007/978-0-8176-4675-2_2,
31. 18 2 Test Functions For the sake of completeness we prove
the following theorem, which is known from analysis (see [7, Sect.
1.8]). Theorem 2.2. For a subset K of Rn the following properties
(a) (c) are equivalent. (a) K is bounded and closed. (b) Every
innite sequence in K has a subsequence that converges to a point of
K. (c) K is compact. Proof. (a) ) (c). We begin by proving that a
cube B D Qn jD1 Ij is compact. Here Ij denotes a closed interval in
R of length l, for every 1jn. Let U be an open cover of B; we
assume that it does not contain a nite cover of B and will show
that this assumption leads to a contradiction. When we bisect a
closed interval I of length l, we obtain I D I.l/ [ I.r/ , where
I.l/ and I.r/ are closed intervals of length l=2. Consider the
cubes of the form B0 D Qn jD1 I0 j , where for every 1jn we have
made a choice I0 j D I.l/ j or I0 j D I .r/ j . Then B equals the
union of the 2n subcubes B0 . If it were possible to cover each of
these by a nite subcollection E of U, the union of these E would be
a nite subcollection of U covering B, in contradiction to the
assumption. We conclude that there is a B0 that is not covered by a
nite subcollection of U. Applying mathematical induction, we thus
obtain a sequence .B.t/ /t2N of cubes with the following
properties: (i) B.1/ D B and B.t/B.t 1/ for every t 2 Z2. (ii) B.t/
D Qn jD1 I.t/ j , where I.t/ j denotes a closed interval of length
2 t l. (iii) B.t/ is not covered by a nite subcollection of U. From
(i) we now have, for every j, I.t/ jI .t 1/ j , that is, the left
endpoints l .t/ j of the I.t/ j , considered as a function of t,
form a monotonically nondecreasing sequence in R. This sequence is
bounded; indeed, l.t/ j 2 I.s/ j when ts. As t ! 1, the sequence
therefore converges to an lj 2 R; we have lj 2 I.s/ j because I.s/
j is closed. Conclusion: the limit point l WD .l1; : : : ; ln/
belongs to B.s/ , for every s 2 N. Because U is a cover of B and l
2 B, there exists a U 2 U for which l 2 U . Since U is open, there
exists an 0 such that x 2 Rn and jxj lj j for all j implies that x
2 U . Choose s 2 N with 2 s. Because l 2 B.s/ , the fact that x 2
B.s/ implies that jxj lj j2 s for all j; therefore x 2 U . As a
consequence, B.s/U , in contradiction to the assumption that B.s/
was not covered by a nite subcollection of U. Now let K be an
arbitrary bounded and closed subset of Rn and U an open cover of K.
Because K is bounded, there exists a closed cube B that contains K.
Because K is closed, the complement C WD Rn n K of K is open. The
collection eU WD U [ fCg covers K and C, and therefore Rn , and
certainly B. In view of the foregoing, B is covered by a nite
subcollection zE of eU. Removing C from zE, we obtain a nite
subcollection E of U; this covers K. Indeed, if x 2 K, there
exists
32. 2 Test Functions 19 U 2 zE with x 2 U . Since U cannot
equal C, we have U 2 E. (c) ) (b). Suppose that .x.j// is an innite
sequence in K that has no subsequence converging in K. This means
that for every x 2 K there exist an .x/0 and an N.x/ for which kx
x.j/k.x/ whenever jN.x/. Let U.x/ D f y 2 K j ky xk.x/ g: The U.x/
with x 2 K form an open cover of K; condition (c) implies the
existence of a nite subset F of K such that for every x 2 K there
is an f 2 F with x 2 U.f /. Let N be the maximum of the N.f / with
f 2 F ; then N is well-dened because F is nite. For every j we nd
that an f 2 F exists with x.j/ 2 U.f /, and therefore jN.f /N .
This is in contradiction to the unboundedness of the indices j. (b)
) (a). Suppose that K satises (b). If K is not bounded, we can nd a
sequence .x.j//j2N with kx.j/kj for all j. There is a subsequence
.x.j.k///k2N that converges and that is therefore bounded, in
contradiction to kx.j.k//kj.k/k for all k. In order to prove that K
is closed, suppose limj!1 x.j/ D x for a sequence .x.j// in K. This
contains a subsequence that converges to a point y 2 K. But the
subsequence also converges to x, and in view of the uniqueness of
limits we conclude that x D y 2 K. The preceding theorem contains
the BolzanoWeierstrass Theorem, which states that every bounded
sequence in Rn has a convergent subsequence; see [7, Theo- rem
1.6.3]. The implication (a) ) (c) is also referred to as the
HeineBorel The- orem; see [7, Theorem 1.8.18]. However, linear
spaces consisting of functions are usually of innite dimension. In
normed linear spaces of innite dimension, com- pact is a much
stronger condition than bounded and closed, while in such spaces
(b) and (c) are still equivalent. As a rst application of
compactness we obtain conditions that guarantee that disjoint
closed sets in Rn possess disjoint open neighborhoods; see Lemma
2.3 be- low and its corollary. To do so, we need some denitions,
which are of independent interest. Introduce the set of sums A C B
of two subsets A and B of Rn by means of A C B WD f a C b j a 2 A;
b 2 B g: (2.2) It is clear that A C B is bounded if A and B are
bounded. Also, A C B is closed whenever A is closed and B compact.
Indeed, suppose that the sequence .cj /j2N in A C B converges in Rn
to c. One then has cj D aj C bj for some aj 2 A, bj 2 B. By the
compactness of B, a subsequence .bj.k//k2N converges to a b 2 B.
Consequently, the sequence with terms aj.k/ D cj.k/ bj.k/ converges
to a WD c b as k ! 1. Because A is closed, a lies in A. The
conclusion is that c 2 A C B. In particular, A C B is compact
whenever A and B are both compact. An example of two closed subsets
A and B of R for which A C B is not closed is the pair A D Z0 and B
D f n C 1=n j n 2 Z2 g. Clearly, A and B are closed
33. 20 2 Test Functions and A C B does not contain any integer.
On the other hand, for every m 2 Z the numbers m C 1=n D .m n/ C .n
C 1=n/ belong to A C B if n 2 Z2 and nm, while m C 1=n converges to
m as n ! 1. Furthermore, the distance d.x; U / from a point x 2 Rn
to a set URn is dened by d.x; U / D inff kx uk j u 2 U g: (2.3)
Note that d.x; U / D 0 if and only if x 2 U , the closure of U in
Rn . The - neighborhood U of U is given by (see Fig. 2.1) U D f x 2
Rn j d.x; U / g: (2.4) Fig. 2.1 Example of a -neighborhood Observe
that x 2 U if and only if a u 2 U exists with kx uk. Using the
notation B.uI / for the open ball of center u and radius , this
gives U D [ u2U B.uI /; which implies that U is an open set. Also,
B.uI / D fug C B.0I /, and therefore U D U C B.0I /: Finally, we
dene U as the set of all x 2 U for which the -neighborhood of x is
contained in U . Note that U equals the complement of .Rn n U / and
that consequently, U is a closed set. Now we are prepared enough to
obtain the following two results on separation of sets. Lemma 2.3.
Let KRn be compact and ARn closed, while KA D ;. Then there exists
0 such that KA D ;. Proof. Assume the negation of the conclusion.
Then there exists an element x.j/ 2 K1=jA1=j , for every j 2 N.
Therefore, one can select y.j/ 2 K and a.j/ 2 A satisfying ky.j/
x.j/k1 j and kx.j/ a.j/k1 j I so ky.j/ a.j/k2 j : By passing to a
subsequence, one may assume that the y.j/ converge to some y 2 K in
view of criterion (b) in Theorem 2.2 for compactness. Hence ka.j/
yk ! 0,
34. 2 Test Functions 21 in other words, a.j/ ! y as j ! 1.
Since A is closed, this leads to y 2 A; therefore y 2 KA, which is
a contradiction. Corollary 2.4. Consider KXRn with K compact and X
open. Then there exists a 00 with the following property. For every
00 there is a compact set C such that KKCCX: Proof. The set A D Rn
n X is closed and KA D ;. On account of Lemma 2.3 there is 00 such
that K30 A D ;. Dene C D KCB.0I /. Then C is compact as the set of
sums of two compact sets; further, CK2; hence CK3K30 . This leads
to CA D ;, and so CX. After this longish intermezzo we next come to
the denition of the space of test functions, one of the most
important notions in the theory. Denition 2.5. Let X be an open
subset of Rn . ForW X ! C the support of , written supp , is dened
as the closure in X of the set of the x 2 X for which .x/ 0. A test
function on X is an innitely differentiable complex-valued func-
tion on X whose support is a compact subset of X. (That is, suppis
a compact subset of Rn and supp X.) The space of all test functions
on X is designated as C1 0 .X/. (The subscript 0 is a reminder of
the fact that the function vanishes on the complement of a compact
subset, and thus in a sense on the largest part of the space.) It
is a straightforward verication that C1 0 .X/ is a linear space
under pointwise addition and multiplication by scalars of
functions. If we extend2 C1 0 .X/ to a function on Rn by means of
the denition .x/ D 0 for x 2 Rn n X, we obtain a C1 function on Rn
. Indeed, Rn equals the union of the open sets Rn n suppand X. On
both these sets we have thatis of class C1 . The support of the
extension equals the original support of . Stated differently, we
may interpret C1 0 .X/ as the space of all2 C1 0 .Rn / with supp X;
with this interpretation we have C 1 0 .U /C1 0 .V / if UV are open
subsets of Rn . In the vast majority of cases the test functions
need only be k times continuously differentiable, with k nite and
sufciently large. To avoid having to keep track of the degree of
differentiability, one prefers to work with C 1 0 rather than the
space Ck 0 of compactly supported Ck functions. The question arises
whether the combination of the requirements compactly supported and
innitely differentiable might not be so restrictive as to be
satised only by the zero function. Indeed, if we were to replace
the requirement thatbe innitely differentiable by the requirement
thatbe analytic, we would obtain only the zero function. Here we
recall that a functionis said to be analytic on X if for every a 2
X,is given by a power series about a that is convergent on some
neighborhood of a. This implies thatis of class C 1 and that the
power series ofabout a equals the Taylor series ofat a.
35. 22 2 Test Functions Furthermore, an open set X in Rn is
said to be connected if X is not the union of two disjoint nonempty
open subsets of X (for more details, refer to [7, Sect. 1.9]).
Lemma 2.6. Let X be a connected open subset of Rn andan analytic
function on X. Then eitherD 0 on X or suppD X. In the latter case
suppis not compact, provided that X is not empty. Proof. Consider
the set U D f x 2 X jD 0 in a neighborhood of x g; this denition
implies that U is open in X. Now select x 2 X n U . Sinceequals its
convergent power series in a neighborhood of x, there exists a
(possibly higher- order) partial derivative of , say , with .x/ 0.
Because is continuous, there is a neighborhood V of x on which
differs from 0. Hence, VX n U , in other words, X n U is open in X.
From the connectivity of X we conclude that either U D X, in which
caseD 0 on X, or U D ;, and in that case suppD X. Next we show that
C 1 0 .X/ is sufciently rich. We fabricate the desired functions
step by step. Lemma 2.7. Dene the function W R ! R by .x/ D e 1 x
for x0 and .x/ D 0 for x0. Then 2 C1 .R/ with .x/0 for x0, and supp
D R0. Proof. The only problem is the differentiability at 0; see
Fig. 2.2. From the power series for the exponential function one
obtains, for every n 2 N, the estimate eyyn n for all y0. Hence .x/
D 1 e1=xn 1=xn D n xn .x0/: This tells us that is differentiable at
0, with 0 .0/ D 0. As regards the higher-order derivatives, we note
that for x0 the function satises the differential equation 0 .x/ D
.x/ x2 : By applying this in the induction step we obtain, with
mathematical induction on k, .k/ .x/ D pk1 x.x/; where the pk are
polynomial functions inductively determined by p0.y/ D 1 and
pkC1.y/ D pk.y/ pk 0 .y/y2 : In particular, pk is of degree 2k and
therefore satises an estimate of the form jpk.y/jc.k/ y2k .y1/:
From this we derive the estimate
36. 2 Test Functions 23 j.k/ .x/jc.k/ n xn 2k .0x1/: If we then
choose n2k C 2, we obtain, with mathematical induction on k, that 2
Ck .R/ and .k/ .0/ D 0. Lemma 2.8. Let 2 C1 .R/ be as in the
preceding lemma. Let a and b 2 R with ab. Dene the function D a;b
by .x/ D a;b.x/ D .x a/ .b x/: One then has 2 C1 .R/ with 0 on a; b
and supp D a; b . Further- more, I./ WD Z R .x/ dx0: The function
Da;b WD 1 I./ has the same properties as (see Fig. 2.2), while R
R.x/ dx D 1. 0 1 2 1 2 1 1 2 2 0.598 Fig. 2.2 Graphs of as in Lemma
2.7 on 0; 1=2 and of 1;2 as in Lemma 2.8, with the scales adjusted
Lemma 2.9. Let aj and bj 2 R with ajbj and deneaj ;bj 2 C1 0 .R/ as
in the preceding lemma, for 1jn. Write x D .x1; : : : ; xn/ 2 Rn .
For a and b 2 Rn , dene the function a;b W Rn ! R by (see Fig. 2.3)
a;b.x/ D nY jD1aj ;bj .xj /: Then we have a;b 2 C 1 .Rn /; a;b0 on
nY jD1aj ; bj; supp a;b D nY jD1 aj ; bj ; Z Rn a;b.x/ dx D 1: For
a complex number c, the notation c0 means that c is a nonnegative
real number. For a complex-valued function f , f0 means that f .x/0
for every x
37. 24 2 Test Functions Fig. 2.3 Graph of . 1;2/;.2;3/ as in
Lemma 2.9 in the domain space of f . If g is another function, one
writes fg or gf if f g0. Corollary 2.10. For every point p 2 Rn and
every neighborhood U of p in Rn there exists a2 C 1 0 .Rn / with
the following properties: (a) 0 and .p/0. (b) supp U . (c) R Rn .x/
dx D 1. By superposition and taking limits of the test functions
thus constructed we ob- tain a wealth of new test functions. For
example, consideras in Corollary 2.10 and set .x/ WD 1 n1x: (2.5)
Further, let f be an arbitrary function in C0.Rn /, the space of
all continuous func- tions on Rn with compact support; these are
easily constructed in abundance. By straightforward generalization
of Lemma 1.6 to Rn , the functions f WD f converge uniformly on Rn
to f , as# 0. The f are test functions, in other words, f 2 C 1 0
.Rn /; (2.6) as one can see from Lemma 2.18 below. Consequently,
for every f 2 C0.Rn / there exists a family of functions in C1 0
.Rn / that converges to f uniformly on compact subsets. We say that
C1 0 .Rn / is dense in C0.Rn /; see Denition 8.3 below for the
general denition of dense sets. Lemma 2.11. For every a 2 Rn and r0
there exists2 C 1 0 .Rn / satisfying supp B.aI 2r/; 01;D 1 on B.aI
r/: Proof. By translation and rescaling we see that it is sufcient
to prove the assertion for a D 0 and r D 1. By Lemma 2.8 we can nd
2 C 1 .R/ such that 0 on 1; 3 and supp D 1; 3 , while I D R 3 1 .x/
dx0. Hence we may write
38. 2 Test Functions 25 .x/ WD 1 I Z 3 x .t/ dt: Then2 C 1 .R/,
01, whileD 1 on 1; 1 andD 0 on 3; 1 . Now set .x/ D .kxk2 / D .x2 1
CC x2 n/. We now review notation that will be needed for Denition
2.13 and Lemma 2.18 below, among other things. In this text we use
the following notation for higher- order derivatives. A multi-index
is a sequence D .1; : : : ; n/ 2 .Z0/n of n nonnegative integers.
The sum jj WD nX jD1 j is called the order of the multi-index . For
every multi-index we write @ x WD @ @x WD @ 1 1 BB @n n ; where @j
WD @ @xj : (2.7) Furthermore, we use the shorthand notation @ D @
@x when we want to differentiate only with respect to the variables
xj . The crux is that the Theorem on the interchangeability of the
order of differentiation (see for instance [7, Theorem 2.7.2]),
which holds for functions sufciently often differen- tiable, allows
us to write every higher-order derivative in the form (2.7); also
refer to the introduction to Chap. 6. Finally, in the case of n D
1, we dene @ as @.1/ . Remark 2.12. In (2.7) we dened the partial
derivatives @ f of arbitrary order of a function f depending on an
arbitrary number of variables. For the kth-order deriva- tives of
the product f g of two functions f and g that are k times
continuously differentiable, we have Leibnizs formula: @ .f g/ D X
! @ f @ g; (2.8) for jj D k. Here D .1; : : : ; n/ and D .1; : : :
; n/ are multi-indices, while means that for every 1jn one has jj .
The n-dimensional binomial coefcients in (2.8) are given by ! WD nY
jD1 j j ! ; where p q ! D p .p q/ q ;
39. 26 2 Test Functions for p and q 2 Z with 0qp. Formula (2.8)
is obtained with mathematical induction on the order k D jj of
differentiation, using Leibnizs rule @j .f g/ D g @j f C f @j g
(2.9) in the induction step. Denition 2.5 is supplemented by the
following, which introduces a notion of convergence in the
innite-dimensional linear space C1 0 .X/: Denition 2.13. Let j and2
C1 0 .X/, for j 2 N and X an open subset of Rn . The sequence .j
/j2N is said to converge toin the space C1 0 .X/ of test functions
as j ! 1, notation lim j!1 j Din C1 0 .X/; if the following two
conditions are both met: (a) there exists a compact subset K of X
such that supp jK for all j; (b) for every multi-index the sequence
.@ j /j2N converges uniformly on X to @ . Observe that the data
above imply that supp K. The notion of convergence introduced in
the denition above is very strong. The stronger the convergence,
the fewer convergent sequences there are, and the more readily a
function dened on C 1 0 .X/ will be continuous. Now we combine
compactness and test functions in order to introduce the useful
technical tool of a partition of unity over a compact set. 2 2 2 3
2 2 1 Fig. 2.4 Example of a partition of unity Denition 2.14. Let K
be a compact subset of an open subset X of Rn and U an open cover
of K. A C 1 0 .X/ partition of unity over K subordinate to U is a
nite sequence 1; : : : ; l 2 C1 0 .X/ with the following properties
(see Fig. 2.4): (i) j0, for every 1jl, and Pl jD1 j1 on X; (ii)
there exists a neighborhood V of K in X with Pl jD1 j .x/ D 1, for
all x 2 V ; (iii) for every j there is a U D U.j/ 2 U for which
supp jU .
40. 2 Test Functions 27 Given a function f on X, write fj D j f
in the notation above. Then we obtain functions fj with compact
support contained in U.j/, while f D Pl jD1 fj on V . Furthermore,
all fj 2 Ck if f 2 C k . In the applications, the U 2 U are small
neighborhoods of points of K with the property that we can reach
certain desired conclusions for functions with support in U . For
example, partitions of unity were used in this way in [7, Theorem
7.6.1] to prove the integral theorems for open sets XRn with C1
boundary. Theorem 2.15. For every compact set K contained in an
open subset X of Rn and every open cover U of K there exists a C 1
0 .X/ partition of unity over K subordi- nate to U. Proof. For
every a 2 K there exists an open set Ua 2 U such that a 2 Ua.
Select ra0 such that B.aI 2ra/UaX. By criterion (c) in Theorem 2.2
for compactness, there exist nitely many a.1/; : : : ; a.l/ such
that K is contained in the union V of the B.a.j/; ra.j //, for 1jl.
Now select the corresponding j 2 C1 0 .X/ as in Lemma 2.11 and set
1 D 1I jC1 D jC1 j Y iD1 .1 i / .1jl/: (2.10) Then the conditions
(i) and (iii) for a C1 0 .X/ partition of unity subordinate to U
are satised by the 1; : : : ; l . The relation j X iD1 i D 1 j Y
iD1 .1 i / (2.11) is trivial for j D 1. If (2.11) is true for jl,
then summing (2.10) and (2.11) yields (2.11) for j C 1.
Consequently (2.11) is valid for j D l, and this implies that the
1; : : : ; l satisfy condition (ii) for a partition of unity with V
as dened above. Corollary 2.16. Let K be a compact subset in Rn .
For every open neighborhood X of K in Rn there exists a2 C 1 0 .Rn
/ with 01, supp X andD 1 on an open neighborhood of K. In
particular, for 0 sufciently small, we can nd such a functionwithD
1 on K . Proof. Consider the open cover fXg of K and let 1; : : : ;
l be a subordinate partition of unity over K as in the preceding
theorem. ThenD P j j satises all requirements. For the second
assertion, apply Corollary 2.4 and the preceding result with K
replaced by C as in the corollary. The functionis said to be a
cut-off function for the compact subset K of Rn . Through
multiplication bywe can replace a function f dened on X by a
function g with compact support contained in X. Here g D f on a
neighborhood of K and g 2 C k if f 2 C k .
41. 28 2 Test Functions We still have to verify the claim in
(2.6); it follows from Lemma 2.18 below. In the case of k equal to
1, another proof will be given in Theorem 11.2. Later on, in
demonstrating Theorem 11.22, we will need an analog of Corollary
2.16 in the case of not necessarily compact sets. To that end, we
derive Lemma 2.19 below. In preparation, we introduce some concepts
that are useful in their own right. Denition 2.17. Let XRn be an
open subset. A function f W X ! C is said to be locally integrable
if for every a 2 X, there exists an open rectangle BX with the
properties that a 2 B and that f is integrable on B. The
characteristic function or indicator function 1U of a subset U of
Rn is de- ned by 1U .x/ D 1 if x 2 U; 1U .x/ D 0 if x 2 Rn n U: U
is said to be measurable if 1U is locally integrable. For the
purposes of this book it will almost invariably be sufcient to
interpret the concept of integrability, as we use it here, in the
sense of Riemann. However, for distributions it is common to work
with Lebesgue integration, which leads to a more comprehensive
theory. Loosely speaking, Lebesgues theory is more powerful than
Riemanns, in the sense that it leads to a process of integration
for more functions and to a simpler treatment of singular behavior
of functions. On the other hand, a thorough treatment of Lebesgue
integration is technically more demanding than that of Riemann
integration. The distinction between the two concepts rarely arises
in the case of the functions that will be encountered in this text.
It is primarily in the description of spaces of all functions
satisfying certain properties that the difference becomes
important. Readers who are not familiar with Lebesgue integration
can nd a way around this by restricting themselves to locally
integrable functions with an absolute value whose improper Riemann
integral exists, and otherwise taking our assertions about Lebesgue
integration for granted. Some of these assertions do not apply to
Riemann integration, but this need not be a reason for serious
concern; we will discuss this issue when the need arises.
Nonetheless, for the benet of readers who are interested in the
relation between the theory of distributions and that of (Lebesgue)
integration we concisely but fairly completely discuss integration
in Chap. 20. In particular, local integrability is intro- duced in
Denition 20.37. Lemma 2.18. Let f be locally integrable on Rn and g
2 Ck 0 .Rn /. Then fg 2 Ck .Rn / and supp .fg/supp f C supp g: Here
supp f C supp g is a closed subset of Rn , compact if f , too, has
compact support; in that case fg 2 C k 0 .Rn /. Proof. We study .f
g/.x/ for x 2 U , where URn is bounded and open. Dene h.x; y/ WD f
.y/ g.x y/. Then the function x 7! h.x; y/ belongs to C k .U / for
every y 2 Rn , because for every multi-index 2 .Z0/n with jjk,
42. 2 Test Functions 29 @ h @x .x; y/ D f .y/ @ g.x y/: Let
B.r/ be a ball about 0 of radius r0 such that supp gB.r/. Then
there exists an r00 with B.r/ C UB.r0 /; furthermore, the
characteristic functionof B.r0 / is integrable on Rn . For every x
2 U the function @h @x .x; / vanishes outside B.r0 /; consequently,
the latter function does not change upon multiplication by . In
addition, we have @ h @x .x; y/ sup x2Rn j@ g.x/j jf .y/j .y/ ..x;
y/ 2 URn /; where jf jis an absolutely integrable function on Rn .
In view of a well-known theorem on changing the order of
differentiation and integration (in the context of Riemann
integration, see [7, Theorem 6.12.4]) we then know that R Rn h.x;
y/ dy is a C k function of x whose derivatives equal the integral
with respect to y of the corresponding derivatives according to x
of the integrand h.x; y/. Furthermore, h.x; y/ D 0 if x 2 U and y
KU , where KU WD .supp f /.U C . supp g//: Now suppose u supp f C
supp g. Then there exists a neighborhood U of u in Rn such that x
supp f C supp g for all x 2 U , because the complement of supp f
Csupp g is open. But this means KU D ;, which implies that .f g/.x/
D 0 for all x 2 U . Lemma 2.19. Let2 C 1 0 .Rn /, 0, R .x/ dx D 1,
and kxk1 if x 2 supp . Suppose that the subset U of Rn is
measurable; see Denition 2.17. Select 0 arbitrarily and dene, for
0, .x/ D 1 n1xand U;WD 1U: Then U;2 C 1 .Rn /; 0U;1; supp U; U:
Finally, U;D 1 on a neighborhood of U . Proof. We have U;2 C 1 .Rn
/ by Lemma 2.18. Because 0, we obtain 0 D 01U1 D 1./ D 1:
Furthermore, if B denotes the -neighborhoodof 0, the support of
U;is contained in supp 1U C supp U C B, and therefore also in the
-neighborhood of U as . The latter conclusion is reached when we
replace U by V D Rn n U ; note that 1 U;D 1 1U D .1 1U / D V;.
43. 30 2 Test Functions Usually in applications of Lemma 2.19,
the set U is either open or closed, but even then its
characteristic function 1U will not always be locally integrable in
the sense of Riemann (see [7, Exercise 6.1]), whereas it is in the
sense of Lebesgue; see Proposition 20.36. The only occasion in the
text where this issue might play a role is in the proof of Theorem
11.22. Finally, there are many situations in which one prefers to
use, instead of2 C1 0 .Rn /, functions like (see Fig. 2.5).x/ Dn.x/
Dn 2 e kxk2 : (2.12) The numerical factor is chosen such that the
integral of over Rn equals 1; thisFig. 2.5 Graph of2, with
different horizontal and vertical scales is the Gaussian density or
the probability density of the normal distribution, with
expectation 0 and variance Z Rn kxk2n.x/ dx D n 2 : (2.13) For
larger values of kxk the values.x/ are so extremely small that in
many situ- ations we may just as well consider as having compact
support. Naturally, this is only relative: if we were to use to
test a function that grows at least like ekxk2 as kxk ! 1, this
would utterly fail. For the sake of completeness we recall the
calculation of In WD Rn.x/ dx. Becausen.x/ D Qn j D11.xj /, we have
In D .I1/n . The change of variables x D r.cos ; sin / in a dense
open subset of R2 now yields
44. 2 Problems 31 I2 D 1Z R2 e .x2 1 Cx2 2 / d.x1; x2/ D 1Z R0
Z e r2 r d dr D 1; or Z R e x2 dx D p : (2.14) We refer to [7,
Exercises 2.73 and 6.15, or 6.41] for other proofs of this
identity. For the computation of (2.13) introduce spherical
coordinates in Rn by x D r! with r0 and ! belonging to the unit
sphere f x 2 Rn j kxk D 1 g in Rn ; see [7, Example 7.4.12] and
(13.37). Next use the substitution r2 D s and formulas (13.30) and
(13.31) below. Problems 2.1.Let URn be a closed set. Prove that the
corresponding distance function satises jd.x; U / d.y; U /jkx yk,
for all x and y 2 Rn . 2.2.Let2 C 1 0 .R/, 0, and 0 supp . Decide
whether the sequence .j /j2N converges to 0 in C 1 0 .R/ if: (i) j
.x/ D j 1 .x j/. (ii) j .x/ D j p .j x/. Here p is a given positive
integer. (iii) j .x/ D e j .j x/. In each of these cases verify
that for every x 2 R and every k 2 Z0, the sequence . .k/ j .x//j2N
converges to 0, and in addition, that in case (i) the convergence
is even uniform on R. 2.3.Letandbe as in Lemma 2.19. Prove that for
every 2 C1 0 .X/, the function converges to in C 1 0 .X/ as# 0.
2.4. Consider 2 C1 0 .R/ with 0 and .x/ D 0 if and only if jxj1.
Further assume that R .x/ dx D 1. Let 01, .x/ D 1.x/, I D 1; 1 R,
and let D 1I. Determine where one has D 0, where 0 1, and where D
1, and in addition, where 0 D 0, 00, and 00, respectively. Now let
.x/ D .x1/ .x2/ for x 2 R2 and let U D II, a square in the plane.
ConsiderD U;as in Lemma 2.19. Prove that .x/ D .x1/ .x2/. Determine
where one hasD 0, or 01, orD 1, and in addition, for j D 1 and 2,
where @jD 0, @j 0, @j 0. Verify that if 01, there is a j such that
@j 0. Prove by the Submersion Theorem (see [7, Theorem 4.5.2]) that
for every 0c1 the level set N.c/ WD f x 2 R2 j .x/ D c g is a C 1
curve in the plane. Is this also true for the boundary of the
support ofand of 1 ? Give a description, as detailed as possible,
of the level curves of , including a sketch. 2.5.For 0, dene 2 C1
.R/ by.x/ D 1pe x2=2 :
45. 32 2 Test Functions Calculate jj . Prove that this function
is analytic on R and examine how closely it approximates the
function jj. Also calculate its derivatives of rst and second
order. See Fig. 2.6. 0.05 0.05 25 0.5 0.5 0.5 Fig. 2.6 Illustration
for Problem 2.5. Graphs of and jjwithD 1=50 2.6. Let U be a proper
open subset of Rn . Let.x/ be as in (2.12) and.x/ D 1 n.1x/, for 0.
Denote the probability of distance to 0 larger than r by .r/ D Z
kxkr.x/ dx: Give an estimate of the r for which .r/10 6 . Prove
thatWD 1U is analytic and that 01. Further prove that .x/.=/ if
d.x; U / D 0; nally, show that .x/1 .=/ if d.x; Rn n U / D 0. 2.7.
Show, for a0, that Z Rn e akxk2=2 dx D 2 a n 2 and Z Rn kxk2 e
akxk2=2 dx D n a 2 a n 2 :
46. Chapter 3 Distributions For an arbitrary linear space E
over C, a C-linear mapping W E ! C is also called a linear form or
linear functional on E. Denition 3.1. Let X be an open subset of Rn
. A distribution on X is a linear form u on C 1 0 .X/ that is also
continuous in the sense that lim j!1 u.j / D u./ as lim j!1 j Din
C1 0 .X/: Phrased differently, in this case continuity means
preservation of convergence of sequences. The space of all
distributions on X is denoted by D0 .X/. The notation derives from
the notation D.X/ used by Schwartz for the space C1 0 .X/ of test
functions equipped with the notion of convergence from Denition
2.13. (The letter D denotes differentiable.) By the linearity of u,
the assertion u.j / ! u./ is equivalent to u.j / D u.j / u./ ! 0,
while j !is equivalent to j! 0. This implies that the continuity of
a linear form u is equivalent to the assertion lim j!1 u. j / D 0
as lim j!1 j D 0 in C1 0 .X/: Example 3.2. We have u 2 D0 .R/ if
u./ D R R .x/ dx, for all2 C1 0 .R/. In- deed, u is well-dened
becauseis continuous with compact support; the linearity of u is
well-known and its continuity can be proved as follows. If limj!1 j
D 0 in C 1 0 .R/, then there exists m0 such that supp j m; m for
all j, while the convergence of the j to the zero function is
uniform on m; m . Therefore we may interchange taking the limit and
integration to obtain lim j!1 u.j / D lim j!1 Z m m j .x/ dx D Z m
m lim j!1 j .x/ dx D Z m m 0 dx D 0: Springer Science+Business
Media, LLC 2010 J.J. Duistermaat and J.A.C. Kolk, Distributions:
Theory and Applications, 33 Cornerstones, DOI
10.1007/978-0-8176-4675-2_3,
47. 34 3 Distributions Example 3.3. We have PV 1 x 2 D0 .R/ if
we dene this linear form, in the notation of Example 1.3, by PV 1 x
W C1 0 .R/ ! C withPV 1 x./ D PV Z R .x/ x dx: Indeed, consider
arbitrary2 C1 0 .R/. Then there exists m0 with supp m; m . We may
write R m m j logjxjj dx DW c.m/0 on account of the conver- gence
of this integral. Hence, (1.3) leads to PV 1 x./ c.m/ sup jxjm j0
.x/j; and this implies that PV 1 x is a continuous linear form on C
1 0 .R/. See Example 5.7 for another proof. For every complex
linear space E with a notion of convergence, it is customary to
denote the space of continuous linear forms on E by E0 ; this space
is also referred to as the topological dual of E. If E is of nite
dimension, every linear form on E is automatically continuous and
E0 is a complex linear space of the same dimension as E. For
function spaces E of innite dimension, this does not apply, and it
is therefore sensible also to require continuity of the linear
forms. This will open up a multitude of conclusions that one could
not obtain otherwise, while the condition remains sufciently weak
to allow a large space of linear forms. Remark 3.4. The study of
general linear spaces E with a notion of convergence is referred to
as functional analysis; this is outside the scope of this text. For
the benet of readers who (justiably) nd the preceding paragraph too
vague, we add some additional clarication. We require that addition
and scalar multiplication in E be continuous with re- spect to the
notion of convergence in E. That is, xj C yj ! x C y and cj xj ! c
x if xj ; x; yj ; y 2 E, cj 2 C, and xj ! x, yj ! y and cj ! c as j
! 1. Fur- thermore, we impose the condition that limits of
convergent sequences be uniquely determined. (The latter is a
consequence of the usual requirement that E be a topo- logical
space having the Hausdorff property, which means that each two
distinct points have nonintersecting neighborhoods.) In that case,
E with its notion of con- vergence is also known as a topological
linear space. A linear mapping u W E ! C is said to be continuous
if u xj! u.x/ as xj ! x in E. If E is of nite dimension, there
exists a basis .ei /1in of E, where n is the dimension of E. The
mapping that assigns to x 2 E the coordinates .x1; : : : ; xn/ 2 Cn
for which x D Pn iD1 xi ei is a linear isomorphism from E to Cn .
The asser- tion is that via this linear isomorphism, convergence in
E is equivalent to the usual coordinatewise convergence in Cn .
Proof. The property that E is a topological linear space
immediately leads to the conclusion that coordinatewise convergence
implies convergence in E. We now prove the converse by mathematical
induction on n.
48. 3 Distributions 35 Let xj ! x in E as j ! 1. Select 1in and
let cj , or c, be the ith coordi- nate of xj , or x, respectively.
Suppose that the complex numbers cj do not converge to c as j ! 1;
we will show that this assumption leads to a contradiction. By
pass- ing to a subsequence if necessary, we can arrange for the
existence of a 0 with cj c for all j. This implies that the
sequence with terms dj D 1= cj cis bounded. Passing to a
subsequence once again if necessary, we can arrange that there is a
d 2 C for which dj ! d in C; here we apply Theorem 2.2.(b). With
re- spect to the E-convergence, this leads to yj WD dj .xj x/ ! d 0
D 0 as j ! 1. On the other hand, for every j, the ith coordinate of
yj equals 1. This means that for every j and k, the vector yj yk
lies in the .n 1/-dimensional linear subspace E0 of E consisting of
the elements of E whose ith coordinates vanish. With respect to the
E-convergence in E0, the yj yk converge to zero if both j and k go
to innity; therefore, by the induction hyp
