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Equation (9) is noteworthy in two other important respects. In the first place, it gives a formula for each at separately, rather than a set of linear algebraic equations for a1,..., an as in the method of collocation, for example. This is due to the orthogonality of the base functions ô1,... ,ôï. Further, the formula for at is independent of n, the number of terms in Sn(x). The practical significance of this is as follows. Suppose that, to obtain a better approximation to f, we desire to use an approximation with more terms, say, k terms, where k > n. It is then unnecessary to recompute the first n coefficients in Sk (x). All that is required is to compute from Eq. (9) the coefficients an+1,..., ak arising from the additional base functions ôï+1,... ,ôê. Of course, if
Chapter 11. Boundary Value Problems and Sturm-Liouville Theory
f, r, and the ôï are complicated functions, it may be necessary to evaluate the integrals numerically.
Now let us suppose that there is an infinite sequence of functions ôõ,... ,ôï,..., which are continuous and orthonormal on the interval 0 < x < 1. Suppose further that, as n increases without bound, the mean square error Rn approaches zero. In this event the infinite series
is said to converge in the mean square sense (or, more simply, in the mean) to f (x). Mean convergence is an essentially different type of convergence than the pointwise convergence considered up to now. A series may converge in the mean without converging at each point. This is plausible geometrically because the area between two curves, which behaves in the same way as the mean square error, may be zero even though the functions are not the same at every point. They may differ on any finite set of points, for example, without affecting the mean square error. It is less obvious, but also true, that even if an infinite series converges at every point, it may not converge in the mean. Indeed, the mean square error may even become unbounded. An example of this phenomenon is given in Problem 4.
Now suppose that we wish to know what class of functions, defined on 0 < x < 1,can be represented as an infinite series of the orthonormal set ô³, i = 1, 2,.... The answer depends on what kind of convergence we require. We say that the set ôð ... ,ôï,... is complete with respect to mean square convergence for a set of functions F, if for each function f in F, the series
f (x) = Yh aiô³ (x), (Þ)
with coefficients given by Eq. (9), converges in the mean. There is a similar definition for completeness with respect to pointwise convergence.
Theorems having to do with the convergence of series such as that in Eq. (10) can now be restated in terms of the idea of completeness. For example, Theorem 11.2.4 can be restated as follows: The eigenfunctions of the Sturm-Liouville problem
— [ p(x)y ] + q (x) y = Xr (x) y, 0 < x < 1, (11)
a1 y(0) + a2 y (0) = 0, bx y(1) + b2/ (1) = 0 (12)
are complete with respect to ordinary pointwise convergence for the set of functions that are continuous on 0 < x < 1, and have a piecewise continuous derivative there.
If pointwise convergence is replaced by mean convergence, Theorem 11.2.4 can be considerably generalized. Before we state such a companion theorem to Theorem
11.2.4, we first define what is meant by a square integrable function. A function f is said to be square integrable on the interval 0 < x < 1 if both f and f2 are integrable11 on
nForthe Riemann integral used in elementary calculus the hypotheses that f and f2 are integrable are independent; that is, there are functions such that f is integrable but f2 is not, and conversely (see Problem 6). A generalized integral, known as the Lebesgue integral, has the property (among others) that if f2 is integrable, then f is also necessarily integrable. The term square integrable came into common use in connection with this type of integration.
11.6 Series of Orthogonal Functions: Mean Convergence
that interval. The following theorem is similar to Theorem 11.2.4 except that it involves mean convergence.
Theorem 11.6.1 The eigenfunctions ô³ of the Sturm-Liouville problem (11), (12) are complete with
respect to mean convergence for the set of functions that are square integrable on
0 < x < 1. In other words, given any square integrable function f, the series (10), whose coefficients are given by Eq. (9), converges to f (x) in the mean square sense.
It is significant that the class of functions specified in Theorem 11.6.1 is very large indeed. The class of square integrable functions contains some functions with many discontinuities, including some kinds of infinite discontinuities, as well as some functions that are not differentiable at any point. All these functions have mean convergent expansions in the eigenfunctions of the boundary value problem (11), (12). However, in many cases these series do not converge pointwise, at least not at every point. Thus mean convergence is more naturally associated with series of orthogonal functions, such as eigenfunctions, than ordinary pointwise convergence.