Some Convergence Proofs in the Vector Analysis of Function Space

Annals of Mathematics

Some Convergence Proofs in the Vector Analysis of Function SpaceAuthor(s): Dunham JacksonSource: Annals of Mathematics, Second Series, Vol. 27, No. 4 (Jun., 1926), pp. 551-567Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1967705 .

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SOME CONVERGENCE PROOFS IN THE VECTOR ANALYSIS OF FUNCTION SPACE.*

BY DUNHAM JACKSON.

1. Introduction. In the space of ordinary geometrical experience, as well as in space of any finite number of dimensions, the formulas of transformation from one system of rectangular coordinate axes to another system having the same origin involve certain sums of N terms each, if X

is the number of dimensions. A familiar notion of recent years is the interpretation of the theory of orthogonal systems of functions as a geometry with infinitely many coordinates. In this geometry a rotation of axes is naturally expressed by means of infinite series, instead of finite sums. For the discussion of the convergence of these series, and others arising out of them, one method, ultimately no doubt the most scientific method, is to set up hypotheses describing as closely as possible exactly those properties that are significant in the demonstrations. But as an introductory exercise there may be some interest in a study of special cases, serving in a way to realize concretely relations which in the abstract have a far wider range of validity. Such a special study is the object of this paper. A presentation of some theorems of convergence will be accompanied by an indication of their bearing on the definition of such concepts as those of gradient, divergence, and curl, in the vector analysis of function space.

2. Properties of a set of orthogonal functions. Let e (x) be a continuous function of period 27r, which is everywhere positive. Let a succession of trigonometric sums Un (x), n = 1, 2, ..., be determined so that

JQ (x)Urn(x)Un(x)dX = 0 (mt+n), f (x)[U(x)]2dx 1,

the order of Un (x) being in each case equal to the greatest integer contained in a-n. The calculation of these sums is equivalent to the formation of a normalized orthogonal sequence of linear combinations of the functions

Fe W) Ve(z) cosx, lte(x) sinx, Vc xcos2x, * a

* Presented to the American Mathematical Society, under other titles, October 25, 1924. and April 11 and December 29, 1925; received in final form March 1926.

5,r51



552 D. JACKSON.

Let ut (x) V Ye (x) U, (x); then

(1) j; um(x)n(x)dx = 0 (m n), f [u(x)]sdx= 1.

The function e (x), being continuous and everywhere positive, has a positive minimum value g. From the second of the equations (1) and the fact that 1/[e(x)] ? 11g, it follows that

J7[Un(x)]2dx 2 J [un(x)]2dx <-7

Hence furthermore, since the integrand is always positive or zero,

J>[Un (X)]P2dx : 1

for 0 ? x ? 27r. Let J. be the constant term of the trigonometric sum [UnWI 2, so that

[ Un (X)P n + = n ()

where v. (x) is a trigonometric sum of order n (at most) without constant term. From the fact that

[Un(x)]2dx ~ 27rcn

it is recognized that 0 ? an :5 1/(2ng), for all values of n. Since

f[U(x)]pdx ==nx+ 0 Tn dx d

it follows that

fJ n (x) dx ! ?nx+ [UnJ(x)]2 dx 2

for 0 <? x 2 7r. But the quantity between bars is itself a trigonometric sum of the nth order, having n (x) for its derivative. By an application of Bernstein's theorem, therefore, I Tn(x) < 2n/q, and

U(X)]2 <_1 2n [Un(-)]2 ! 2ig + < C, n, Un(x) ? VC~n,



VECTOR ANALYSIS OF FUNCTION SPACE. 553

where C1 is a quantity independent of n. The maximum of e (x) being denoted by G,

I Un Wz = Ve(x) Un (x) ?_VG C1 n,

or, if VG C1 C2, a number independent of n,

Un(x) I < C2 Vn for all values of n and x.

3. Theorems on the development of given functions in series of orthogonal functions. Let f(x) be a given continuous function of period 27r. The following facts are well known, and readily verified:

Among all expressions of the form

it

Son(x) = akuk(X), k=1

for a given value of n, there is just one for which the integral

[fx) - Son (x)]2 dx

is a minimum, the minimizing coefficients being given by the formula

42n (2) ak = f(t) Ik(t) d t.

When the coefficients are thus determined, the corresponding sum n W(x) being denoted by Sn (x),

f[f(x)-sn(x)]2 d x - [f(x)]2 dx- ak,

from which it follows (since the left-hand member is non-negative) that

n 271

a2 < f[f(x)12 dx k=1

for all values of n, and that the infinite series ak is convergent. The sum of the series is equal to the integral on the right. For

4f(X) _SJ(12 (x)2 fx) - in(x) d2 f~f~x) ~ ~d- Jo [1Vex

dx,



554 D. JACKSON.

where Sn (X) _

Si(X) VQ(x) _ ak Uk(X) Ye (W) k=1

is a trigonometric sum; by Weierstrass's theorem, applied to the continuous function F(x) = f(x)/lV (x), there are trigonometric sums Tn (x) for which F(.x)-Tn(x) is uniformly small, so that

P2n

Jef (x) [F(x) -Tn (x)] 2 dx

approaches zero with increasing n; and the corresponding integral with Sn (x) in place of Tn (x) will approach zero with equal or greater rapidity, by reason of the minimizing property of the coefficients that define S&(x).

If g(x) is another function satisfying the conditions imposed on f(x), and

n

anr (X) = z bk Uk (X) k=1

the corresponding linear combination of the u's,

[f(x) -st, (x)] [g (x)-rnt (x)] dx J =gof(x)g(x)dx- zakbk 0 ~~~~~~~~~~~k=1 by Schwarz's inequality,

[f(x) - Sn (X)] [(x) W- n (X)] d

go [f(x) - Sn (x)]dx [g (x)- an (x)]2 dx;

as each factor on the right approaches zero when n increases indefinitely, the left-hand member does the same, and

ak bk = ff(x)g(x)dx. k=1

Finally, if the series I ak Uk (x) for a given function f(x) is uniformly convergent, it has f(x) for its sum. For if the sum of the series is i(x), and if f(x)-i (x) co(x), then

Yk -fo LO(X)Uk(x)dx - 0, [f o ([)1 d 0, k=1k

and hence &o(x) 0 identically.




4. An inequality for the coefficients. Let it be assumed hence- forth, unless otherwise stated, that the functions e (x) and f(x) have continuous second derivatives for all values of x. Let M be a common upper bound for |f(x)I, If(x)j, and Jf"(x)!, and, if el(x) 1/1V(x), let G, be a common upper bound for e,(x), |Q (x)j, and jle'(x) . Let F(x) have the same meaning as in the preceding section. Then

I F"1(z) I - It da [f(x) el(x)] if elt Q+2f' 1el+ fQ1' 4 GM.

When n is a positive even integer, let m n- 1; and when n is odd, let mr- (n- 1). Since F'(x) satisfies a Lipschitz condition with coefficient 4GI M, there exists for each positive integral Value of m (and so for n ? 3) a trigonometric sum Tn(x), of the rnth order at most, such that

JG, M CS M IiF(x) -- Tn (x) _ 2 i - M - n2

where* K is an absolute constant (independent of m, n, x, f(.x), and Q(x)); in the last member, which is explicit enough for the following work, C3 is independent of f(x), though it depends on Q(x) through the factor Gl.

The number of terms in T (x), when sines and cosines are counted separately, is at most 2m +I1 < n. Each of these terms can be expressed as a linear combination of some or all of the functions Uk(x), k 1, 2, ..*, n. So tit(x) - e(x) Tn(x) is a linear combination of ul(x), u., Un(x), and

27 [f(X)- tn()]2dx = x) [F(x)- T,&(x)Pdx

4M2

where C4, like the other C's to be used below, is independent of f(x), as well as of n and x. By the minimizing property of the coefficients in st(x) it follows then that'

n g27 [if (x) -Sn (X)] 2dx < 4.n,

In consequence of the relations cited in the preceding section,

2- o n 00

[f (x)-s,,(x)] dx - fka1 __ a 2 a 2

* Cf., e. g., D. Jackson, On approximation by trigonometric sums and polynomials, Trans. Amer. Math. Soc., vol. 13 (1912), pp. 491-515; see p. 496, Theorem III.



556 D. JACKSON.

Hence

a2 < I a ? 4 V _<< 05M _+

~~ n4

7 n - n 2

< (n +1)2

The argument has involved a supposition that ?i ? 3, n + 1 > 4. But it is clear from the definition (2) that a relation of similar form, possibly with a different factor in place of C5, can be written down for a,, a2 and a8 (and could have been written down immediately, without reference to the reasoning of the present section, for any finite number of the coefficients). So it can be asserted that there is a constant C, independent of f(x), though depending on e (x), such that

CM I anz I Z n2

for all values of n from n 1 on. With the concluding result of ? 2, this implies incidentally that

a un (x) I < Cs M/ n3/2, and hence that* the series ak Uk (x) converges uniformly to the value f(x).

5. Equations of transformation connecting two orthogonal systems. Let a(x) be another function which like &(x), has the period 27r, a continuous second derivative, and a positive minimum. Let V1 (x), v2 (x), * , be the normalized orthogonal system corresponding to a(x), as the system u, (x), u2 (x), . * *, corresponds to e (x). Each function vk (x), being equal to a trigonometric sum multiplied by V1/?(x), has a continuous second derivative, and therefore can be expanded in a uniformly convergent series of the form

(3) Vk (x) E lSkihi (X). i

For each value of k, in consequence of the facts summarized in ? 3,

Pg ?o [vkx(z2dx 1, and if k $ 1

Pki~ r~ ,l ,

Vk~ W nWdx~0

* Cf. D. Jackson, Note on the convergence of weighted trigonometic series, Bull. Amer. Math. Soc., vol. 29 (1923), pp. 259-263; W. Stekloff, Sur une application de la th6orie de fermeture au probleme du d6veloppement des fonctions arbitraires en series procedant suivant les polynomes de Tch6bicheff, M6moires de l'Acad. Imp. des Sci. Petrograd, ser. 8, Classe phys. math., vol. 33 (1914), No. 8.




Similarly, expansion of the u's in terms of the v's leads to relations of the form

(4) ui (X) W qik Vk (X), ijk 1, Eqikqjk = 0 (i$j). k k k

Here the coefficient qik in the expansion of t (x) in terms of the v's is given by the formula

qik fo Ui (X) Vk (x) dx;

as this integral is the same as the one that defines pki,

qik Ph.

and the last two equations of (4) can be rewritten in the form

(5) p2= 1, P ki Pkj 0 (itj).

There is of course nothing new in the form of these relations; the point is that their validity is assured by the hypotheses that have been adopted.

6. Equations of transformation for vectors and for tensors of the second rank. Let f(x) be a continuous function of period 27i (the hypothesis with regard to the existence of f" (x) being for the moment superfluous), and let

ai J f (x) ui (x) dx, bk f f (x) vk (x) dx.

Since the series expressing Vk () in terms of the u's is uniformly convergent, and remains so after multiplication by f (x),

(6) bk f(X) [PkiUi(x)] dx = zpkiai.

The function f(x) may be regarded as constituting a vector in function space; the equations (3), for the successive values of k, are the equations of transformation of a fundamental system of unit vectors; the a's and the b's measure the components of f(x), as resolved according to the two fundamental systems; and the identity of the coefficients in (6) with those in (3) shows that the components of a (more or less) arbitrary vector are transformed like the constituent vectors of a fundamental system, as would be expected by analogy from space of a finite number of dimensions.



558 D. JACKSON.

As a result of ? 4, applied to the expansion (3), 1 p?j I _ ck/i2, where Ck is independent of i. By ? 2, l u(x) ? < C2Vi, so that Ipkiui(X) ? < CkiI/ Hence the series

Vk(X) - 2pkiUi (X), vi((y) i j

are absolutely convergent, and can be multiplied together:

(7) Vk (x) Vj (y) - 2 ,iPkiPljUi (x) Uj (Y). i J

For the sake of an application to be made presently, it is to be noted that the double series is uniformly as well as absolutely convergent. The identity (7), considered for k 1 1, 2, ,.., I 1, 2, *.. , may be regarded as defining the equations of transformation of a tensor of the second rank.

Let f(x,. y) now be a function of two variables, continuous with regard to both together, and of period 27v in each. Let

aij f f(x, y) ui (x) uj (y) d x d y,

2in

bk1 = f27E f(X, y) Vk (x) v (y) dx dy.

If the integral defining bk1 is evaluated by means of (7), it is found that

bkl = L ' Pkijpljaij,

which may be taken to mean that the doubly infinite matrix of the quantities aj has tensor character.

It is to be shown below that certain differential operators follow the same equations of transformation.

7. Analogue of the curl. As a first illustration, let

(8) (DW) p(y), y =f(x),

where f(x) is a function of period 27r, with a continuous second derivative, and So (y) has a continuous first derivative for all values of y. A function of x may be typified in function space either by a point or by a vector. It is convenient here to think of f(x) as a point, and of ED (x) as a vector. Then the definition (8) makes ED a vector point function in function space.

Let at f(x) Us (x) dx, bk = Jf(x) Vk (x) dx,

(9) =~ JE (x) (x) d, Bk JD (x) V (x) dx

Ai-J O (x) ui (x) dx, Bk - o (X) Vk (x) dx.




It is known that f(x) a mj (x) bk Vk (X),

i ~~~~k

the series being uniformly convergent. Through the dependence of (D on f (x), each of the quantities Ai, Bk is a function of the infinitely many variables a,, a2, * * *, and likewise of the set of variables b,, b2. * * *. Such a partial derivative as aAda aj can be interpreted immediately, without further reference to the technical theory of functions of infinitely many variables. The requirement of the continuity off" (x), to be sure, restricts the a's and the b's to an extent that is not readily specified in terms of these variables themselves. (The necessary condition I aij I CM/i2, of ? 4, is not sufficient.) But if the a's have been given a set of admissible values, the replacement of aj by aj + Aay, for the sake of defining a corresponding partial derivative, does not impair the continuity of the second derivative of the sum of the series, as it has the effect merely of adding the single term Aajuj(x). In fact, when x, a1, a2, ... , are regarded as independent variables, and similarly x, bl, b2, . .,it is recognized at once that

f(x) - uj(), a f() -VI, a ajb a P(X) - Sp, ) Ujz) Wt WPX '(y SWVI

In the integral defining Ai, furthermore, the conditions for differentiating with regard to aj under the sign of integration are satisfied, the integrand being for the purpose in hand essentially a function of the two variables aj and x. The result is that

(1)aAi u~ a Wxd (lo.) i= ui~) W O ) dx =a /'[f (x)] Ui(x) utj(x) dx.

Similarly, aBk

(11) abi fiJ [f (] Vk (X) vl(X) dx.

Finally, Vk VI has the uniformly convergent expansion

Vk(X) VI (X) - Z jjPki plj Ui (X) Uj(X)e i J

On substitution in (11) and comparison with (10), this gives

aBk v v jaAi (12) =l 2: . : i pbPl aa.

The matrix of the quantities AM/a aj thus has the character of a tensor.



560 D. JACKSON.

A similar calculation can be carried through if the definition of 41 is replaced by d(x) =- s (y')

- S[f, (X),

the function So having the same character as before, while e(x), 6(X),

and f(x) are supposed now to possess continuous third derivatives. In the first place, it can be shown by reasoning analogous to that of ? 4, and by reference to a corresponding theorem on trigonometric approximation, that ?a.I< k/n3, where k1 is independent of n. On the other hand, Bernstein's theorem, applied to the relation U. (x)I < 1C/7n1/2, shows that I U (x) I < VFC1n3I2, whence it follows further that I un (x) I ? k2 n812 anu' (x) ? k3/n812. So f' (x) is represented by the uniformly convergent

series i ai u' (x). In the same way, f' (x) kbk vk (x). Also, the representation of v, (x) in the form j plj uj (x) may be differentiated term by term for each value of 1, yielding a series which is absolutely-uniformly convergent in each case. Hence

Vk (X) VZ (X) pki Plj Ui (X) Uj(X), ii

still with uniform convergence of the series of absolute values. The calculation then follows the same lines as before, the relations being

aj f' (X) - (x), a f'(x) = b Ha IPx-E fy)u~) as q)(x)-o' (y') v'(x), aj a b

a r (x)] iuj(x) uj(x) dx, aaj o

ab = 9[ (X)] Vk (X) v (x) d x = 7 ki PV a Ai

If W (x), more generally, has the form

W(X) 5= (Xy y'), y _ f(x), where 9p has a continuous derivative with regard to each argument, while f(x)ye (x), di (x) are as in the preceding paragraph, the changes introduced are illustrated by the relations

a W ( ) = sy (X y, y') uIj(x) + sy' (x, yY') u,(x),

ai 27_ Spy (x, y, y') ui (x) Uj (x) d x + sof , (x,y, y') zti (x) uj (x) dx,

and (12) is verified again.




It is clear that higher derivatives may be admitted without causing any new difficulty, provided that e (x), o(x), and f(x) have a sufficient number of continuous derivatives.

A more elaborate example, designed for a purpose which will appear in the next section, is

(13) ( I D(x) f kK(x, t) 4f (t)] d t,

where the function 9p has for the moment a continuous first derivative with regard to its argument, f(t) is of period 2 n and has a continuous second derivative, and K(x, t), of period 27r with regard to each variable, is continuous in both together. The quantities aj, bk, Ai, Bk are defined again by the formulas (9), interpreted now in terms of the present definition of d. The relations

(14) a @(z)-= nK(x, t) St'[f(t)] uj(t) dt, 8aaj (14) ao

aAi - 12zg2 K(x, t) So' [f(t)] ui(x) uj(t) dl dx, Aaj e o

together with the corresponding expressions for a D/a b1 and a Bk/a bl, and the uniformly convergent development

Vk (X) VI(t) = 2:fki plj U (X) Uj(t),

lead to (12) once more as the general equation of transformation. The more general form

C (x) = J K(x, I) S (t, y, y') dt, y =- f I

or a still more complicated one involving higher derivatives, can be dis- cussed in the same way, if f(x), e (x), and a (x) are sufficiently differentiable.

The recurring tensor equation for the transformation of a Al/a aj, though its verification depends each time on the explicit form of the vector point function, or functional, to which it is applied, indicates that the matrix of the partial derivatives has an invariant character that may reasonably be supposed to have some ulterior significance. It generalizes the curl of three-dimensional vector analysis. The analogy is still more apparent in the skew-symmetric tensor having (a AAa aj)- (a A/a aj) as its general element. The transformation

36



562 D. JACKSON.

aBk aB? ' aAi a Aj a bi abk i \ aa }aaI

may be deduced from (12) at once, by suitable manipulation of the subscripts. 8. Analogue of the divergence. The developments of the preceding

section inevitably suggest consideration of the series

(15) ai a ai '

which results from "contraction" of the derivative tensor. Here the laying of the analytical foundation is at first not so facile. For if 0 (x) is taken identical with f(x), to give the simplest imaginable non-trivial case, aAi/lai = 1, and the series (15) is divergent. If 0d(x) =- sp[f(x)],

aAi f27y [f(x)] Eui (x)]Jdx,

which does not approach zero, in general, as i increases; when e (x) 1, for example, the definition of the it's will naturally be so arranged that ui (x) = (1/}Tr) cosjx or (1/1V7_) sinjx, for i > 1, with a suitable relation between j and i, and then

lim = i = lim [O 9 Lf(x)] (l A cos 2jx)dx = 21 so' U(x)] dx,

which is in general not zero. The definition (13), however, leads to something constructive once more.

Let it be supposed now that so has a continuous third derivative, and, for simplicity of statement, that K(x, t) has continuous partial derivatives of the first four orders, though not all of these will be used; the functionf(t), which serves as independent variable in the definition of the functional O (x), is to have merely a continuous second derivative, as before. Let

i (t) = K(x, t) ui (x) dx.

For each value of t, the quantity ti (t) is the ith coefficient in the expansion of K(x, t) in series of the functions u2 (x). As K(x, t), Kx(x, t), Kxx (x, t) are supposed periodic in each variable and continuous in both, they are bounded. If M, is a common upper bound for their absolute values, it follows from ? 4 that

I Vi W I :5CM




the factor C is independent of the form of K(x, t) as a function of x, and so, in particular, independent of t. Furthermore,

Vi4 (t) JoKt (x, t) u, (x) dx, qi' (t) JoKtt (x, t) ui (x) dx.

If M2 is a common upper bound for the absolute values of the nine functions K, Kx, Ken, Kt, Ktx, Kts, Ktt, Ktto Ktt.,x it appears more comprehensively that

CM CM2 CM2

for all values of t and i. From (14),

a Ai f2 [fo(t)] pi (t) uj (t) dit.

Since gS has a continuous third derivative with regard to its argument, while f(t) has a continuous second derivative, '[f(t)] has a continuous second derivative with regard to t. If Al8 is a common upper bound for the absolute values of So' [f(t)] and of its first and second derivatives with regard to t, and if 'Uf(t)] fi (t) = i (t), then

coi(t) < iCM2 M

d 2'()IM? CM.M X @' Vt i Wt d(so' [fJ It] + I Sp, [AM (t)] go I e

| A () <4 CM12 Ms

So 4 CM2 M8/i2 is a common upper bound for ooi (t) wA(t) 1, and IWa'(t)I. Consequently, by another application of ? 4,

I o (t) uj (t)dt ? j2J2s 2

where the essential property of Q is merely that it is independent of i and j. The last relation shows incidentally, of course, that a Ad a ai I < Q/i4y

and so that the series (15) is absolutely convergent. But further than that, it means that a Aid a aj is the general term of an absolutely convergent double series. On the other hand, by Schwarz's inequality,

36*



564 D. JACKSON.

(lk rJi) ? ( = 1,

.LrPkirPkjI 1, k

for any values of i and j. So a DAi < a DAi

k Jki~kj a aj aAj zlktk aa; I-I aaj

Hence it appears that the triple series

2: Y 2: Pki Pkj aA i j k aaj

is absolutely convergent. By (12), the double sum with regard to i andj, for fixed k, is equal to a Bk/abk. So the value of the triple series is

a aBk k abk

which is thus seen to be convergent. By (5), on the other hand, the sum with regard to k, for fixed i and j, is 0 if j t i, and is aAi/laa if j i, whereby the value of the triple series becomes 2i (a A/a at). For the case in hand, then, the quantity (15) is defined, and is invariant under transformation of axes:

v aBk a Ai k abk - aaI

It corresponds to the divergence in ordinary vector analysis. It is naturally desirable to express this quantity in a form not explicitly

dependent on any particular orthogonal system. The invariance being once established, such an expression is readily obtained. It is sufficient to carry through the calculation for the case Q(x) 1, with functions ui(x) defined by the equations

Ui(x) >2-' ts(z)- 1 - ' = Dcosjx __ i 1); Ui W ~~U2j (X) ,u2.j41 (X) --i7

by the demonstration just completed, any other weight function e(x) satisfying the hypotheses would necessarily lead to the same result.

In general (cf. (14)),

aAti e27rr27SK(x, t)?S'[f(t)]ui(x)ui(t)dt dx. a ai Jo




In the simple case under discussion,

a AA t 1 2;T2%K(x, t) p' [f(t)]dt dx, aa1L 2 o o

.aA2j+ aA2j+= - 2n JJ2 K(x t0s' [f (t)] cosj (x- t) dt dx.

aa2j aa2j+l Tr o o

All the functions involved being continuous, the successive integrations may be performed in either order. By integration by parts,

fo K(x, t) cosj(x- t)dx =- K (x, t) sinj(x - t) dx;

the absence of another term on the right is due to the fact that K(x, t) and sinj(x -t) have the period 2ir with regard to x. Hence

2 Ai= xA1 1 zgr K (x, t)S[f(t)]sinJ(x t)dxdt i a a ala ir j OS O

The series 2j [sinj(x - t)]/j is convergent for all values of x and t. For pairs of values (x, t) belonging to the square 0 ? x ? 2r, 0 t _ 27r, the function

2(x t _ sinj(x t)

may be alternatively defined by the equations

WI (xI t) 4 (t -x + 7r) when x> t

(16) Z(x, t)= 4(t-x-a) when x <t,

WIf(x, t) 0 when x t.

The series is not uniformly convergent over the square. But it is uniformly convergent over any closed sub-region of the square not containing a point of the line x = t; and its partial sums are uniformly bounded everywhere.* So it is readily seen to be integrable term by term, even without reference to theorems of the highest generality on the integration of non-uniformly convergent series. The properties justifying the termwise integration are not affected when the series is multiplied by the continuous factor Kx (x, t) p [f(t)]. Consequently

* Cf., e. g., D. Jackson, tYber eine trigonometrische Summe, Rend. Circ. Mat. di Palermo, vol. 32 (1911), pp. 257-262.



566 D. JACKSON.

a Ani I e K(x, t) p'[f(t)] dxdt X a ai 27 040

-I f2n ;2n K (x, t) sX'jf(t)] IF(x, t) dxdt.

When l(x, t) is regarded as defined by the equations (16), the right-hand member is intelligible without reference to any particular system of orthogonal functions; it constitutes an invariant definition of the "divergence" for the case in hand.

The substitution of o (t, y, y') for So (y) in the definition of W (x) introduces no essential difficulty, though the formulas, including the invariant formula for the divergence, are correspondingly longer. There is occasion incidentally to find an upper bound for the absolute value of

W i (t) u5j (t) dt,

where 0i (t) Soy (t, y, y') ti (t); but as an integration by parts gives

J2 (t) uj (t) d t - (t) uj (t) d t,

because of the periodicity of the functions, the order of magnitude of the result is seen to be a question merely of the existence of a sufficient number of continuous derivatives.

9. Analogue of the gradient. The same order of ideas naturally gives rise to a concept to which the name gradient can be applied. But this concept turns out to be long since familiar under another guise, in the case to which one first thinks of applying it, and more recently recognized of the form appropriate to the present discussion.*

Let

S =J9(t, y, y') dt, y = f(t)

where So and f have a sufficient number of continuous derivatives, and f(t) has the period 27r. (It will be more than sufficient if all the derivatives through the fourth order are continuous.) Considered as a functional with f(x) as its argument, this a) does not depend on x, and has the character, not of a vector, but of a scalar point function in function space. When the functmons ui(x) are looked upon as a system of mutually orthogonal

* Cf. P. Levy, Lqeons d'analyse fonctionnelle, Paris, 1922, pp. 127-128.




unit vectors, the gradient of d) is defined by formal analogy as the sum of the series

(17) Ui (X). iaai Now

a aj - 2 f[?u (t, y, y') ui(t) + sy (t, y, y') UZ(t)] dst

T- soy5? (t, y, y') uj (t) d t -J d t [Soy, (t, y, y')] uti (t) d t,

the last form resulting from an integration by parts. So a 0/a ai is the ith coefficient in the expansion of

(18) py(x, Y, y') d spy, (X, Y, ) dx

in series of the u's. The hypotheses being sufficient to insure convergence, the sum of the series is the "functional derivative" (18). As the form of (18) is independent of the choice of the system of u's, no further proof of invariance is needed.

The character of (18) as a gradient is more forcibly brought out by the statement, which may be verified without difficulty, that the directional derivative of (D in any direction (in the space of the class of functions considered) is equal to the magnitude of the vector (18), multiplied by the cosine of the angle between this vector and the direction in which the derivative is taken, the geometrical terms being of course understood in the light of the definitions appropriate to function space.

THE UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MINN.



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Some Convergence Proofs in the Vector Analysis of Function Space