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

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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.

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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.