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Ceshro-One Summability and Uniform Convergence of Solutions of a Sturm-Liouville System
D. H. Tucker University of Utah, Salt Lake City, Utah 84112
And
R. S. Baty Sandia National Laboratories, Albuquerque, New Mexico 871 85-0825
Introduction
Galerkin methods are used in separable Hilbert spaces to construct and compute
L2[0,n] solutions to large classes of differential equations. In this note a Galerkin
method is used to construct series solutions of a nonhomogeneous S turm-Liouville
problem defined on [O,n] . The series constructed are shown to converge to a specified
du Bois-Reymond function f in L2 [O,n] . It is then shown that the series solutions can
be made to converge uniformly to the specified du Bois-Reymond function when
averaged by the Cesho-one summability method. Therefore, in the Cesho-one sense,
every continuous function f on [O,n] is the uniform limit of solutions of
nonhomogeneous S turn-Liouville problems.
CesBro-One Summability and the Solutions of a Sturm-Liouville System
Consider the regular S turm-Liouville problem
defined on [O,n]. This problem has an associated complete orthonormal system of
eigenvalues and eigenfunctions (A,, q,, ) in L2 [0, n] , where
1
DISC LA1 MER
This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
DISCLAIMER
Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.
an = I - n 2
qn = sin nx.
Also, consider a du Bois-Reymond function f on [O,n]. This means that f is ca
continuous, and f has a Fourier series f - zanqn , where a,, = e f , q , > such that n=l
ca
the Fourier series canqn diverges on a dense G, in [O,n] , du Bois-Reymond [ 11.
Assume further that f(0) = f ( n ) = 0 . From a theorem of FejCr [2], it is well known that
the Fourier series of f converges uniformly after an application of Cesho-one
summability ((C,l) summability). A modern outline of this result may be found in
Bruckner et al. [3], whereas a detailed development of Ces&ro summability is given in
Hardy [4].
n=l
N For each positive integer N , set g = Inan pn , then there exists a solution y N to
n=l
the nonhomogeneous S turm-Liouville problem
namely,
The function y N diverges on a dense G, in [O,n] , but converges uniformly to f when
averaged by the (C,1) summability method. Thus, in the sense described here, f is a
uniform limit of solutions to the problem
where the right-hand side is quite badly behaved as N becomes large. In case the Fourier
series for f actually does converge uniformly, the arguments still apply and f is a
uniform limit of solutions to the Sturm-Liouville problem in the same sense.
2
Moreover, notice that for any fixed value of N , the solution y N of the Sturm-
Liouville problem (3) appears to be well behaved. It is only as N increases without
bound that { y , } becomes pathological. The (C, 1) summability method applied here
provides a “canonical regularization” of the sequence of solutions { y , } by assigning it to
an averaged sequence where the limiting value converges uniformly to f . From this
example, it is suspected that more general summability methods may have significant
application in regularizing solutions of differential equations modeling complex multi-
scale physical phenomena, such as turbulent flow. In the next section, the Sturm-
Liouville problem (1) is derived from a simple heat transfer problem with continuous
initial data.
The Sturm-Liouville System via a Heat Equation
To motivate the Sturm-Liouville system (I), consider the following initial-boundary
value problem from the theory of heat transfer:
where f is a du Bois-Reymond function. Separating variables
one obtains
where 2 a, = M
v,, = sin ux ,
Equation (8) also implies that
z, =exp{A,t}
sothat u becomes
(9)
3
Notice that evaluating equation (1 1) at t = 0 yields
where z, (0) = a, = < f , qn >, then since f is a du Bois-Reymond function, the series m
converges in L2[O,n] to f . However, xanqn diverges on a dense G, in n=l
2 a n q n n=l
[O,n] and after the application of (C,1) summability, converges uniformly to u(0,x).
N auN at n=l
N
Next, set uN ( t ,x) = Can~,qn SO that - = &,Anznqn, where n=l
= ~ a n A n q n = g is the g, defined in the previous section. Furthermore,
of equation (4) is the solution to the
ordinary differential equation + uN I,zo = y i + y N = g N of equation (5).
is pathological at t = 0, it
converges absolutely and uniformly on (O,n]. This can be shown as follows. Since
au at
Finally, notice that while the series defining -
{a, }E Z 2 , the {a, } are bounded by some constant M ; then
m
Since t is fixed, the integral test J(1- x2)exp{ (1 - x2) t }dx < 00 implies that the right- 1
hand side of equation (1 3) is finite, as required.
4
Conclusions
The following conclusions may be drawn from the above example:
Cl . In the (C,1) sense, every continuous function g on [O,n], which satisfies
g(0) = g(n) and can be mapped to an f satisfying f(0) = f (n) = 0 via a linear
transformation, is a solution to the Sturm-Liouville problem given by system (3).
C2. The method used for approximating the solution is of the Galerkin type. The
solutions y N are projections into the finite dimensional subspaces of the Hilbert
space L2 [0, n] , which have basis elements { qn
C3. The Galerkin method fails to work in the sense that the resulting approximate
solutions do not converge to a pointwise solution on [O,nJ (although L2
convergence is assured). Furthermore, the functions f E C[O, n] , whose Fourier
series converge pointwise on a dense G, subset of [O,n], is of the first Baire
category in C[O,n] .
.
Open Research Problems
Suppose {pn } n E q is an orthonormal basis for the Hilbert space H = L2[a,b]. Let
f E H , then f 2 (ar,,) E I;, where a,, = 5 f (t)q,, (t)dt and by the Riesz-Fischer theorem
[3] z is an isometric isomorphism. The following research problems are raised by
consideration of the above discussion:
a
m
P1. Find necessary and sufficient conditions on {a,} such that zanyn converges n=l
uniformly to f on [a,b] .
P2. Find summability methods S such that {S(a,)} satisfies the conditions of P1.
P3. Do the results of P1 and P2 extend to arbitrary H and 18 ?
5
References
1. du Bois-Reymond, P., 1876, “Untersuchungen uber die Convergenz und Divergenz
der Fourier’schen Darstellungsfomeln,” Munchen Ak. Abh., 12, pp. 1-102.
Fejkr,, L., 1904, “Untersuchungen uber Fouriersche Reihen,” Math. Ann. 58, pp.51-
69.
Bruckner, A. M., Bruckner, J. B. and Thomson, B. S., 1997, Real Analysis,
Prentice-Hall.
Hardy, G. H., 1949, Divergent Series, Oxford University Press.
2.
3.
4.
Acknowledgements
Sandia in a multiprogram laboratory operated by Sandia Corporation, a Lockheed
Martin Company, for the United States Department of Energy under contract DE-AC04-
94AL8 5000.