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The theory of Fourier series discussed in Chapter 10 is just a special case of the general theory of Sturm-Liouville problems. For instance, the functions
ôï (x) = V2sin nn x (13)
are the normalized eigenfunctions of the Sturm-Liouville problem
/ + Xy = 0, y(0) = 0, y(1) = 0. (14)
Thus, if f is a given square integrable function on 0 < x < 1, then according to Theorem 11.6.1, the series
f (x) = Yh bm ôø (x) = ^2Yh bm sin mn x’ (15)
m=1 ø= 1
bm = f f ^)ôø (x) dx = V2 f f (x) sin mn x dx, (16)
converges in the mean. The series (15) is precisely the Fourier sine series discussed in Section 10.4. If f satisfies the further conditions stated in Theorem 11.2.4, then this series converges pointwise as well as in the mean. Similarly, a Fourier cosine series is associated with the Sturm-Liouville problem
/ + xy = 0, Ó (0) = 0, y (1) = 0. (17)
Let f (x) = 1 for 0 < x < 1. Expand f (x) using the eigenfunctions (13) and discuss EXAMPLE the pointwise and mean square convergence of the resulting series.
1 The series has the form (15) and its coefficients bm are given by Eq. (16). Thus
ã f1 V2
bm = v 2 sin mnx dx =-(1 — cos mn) (18)
m ' mn
Chapter 11. Boundary Value Problems and Sturm-Liouville Theory
and the nth partial sum of the series is
1 — cos mn .
S (x) = 2 Ó ------------------sin mnx.
The mean square error is then
[ f(x) — Sn(x)]2 dx.
By calculating Rn for several values of n and plotting the results, we obtain Figure
11.6.2. This figure indicates that Rn steadily decreases as n increases. Of course, Theorem 11.6.1 asserts that Rn ^ 0 as n ^æ. Pointwise, we know that Sn(x) ^ f (x) = 1 as n ^<x>; further, Sn(x) has the value zero for x = 0 or x = 1 for every n. Although the series converges pointwise for each value of x, the least upper bound of the error does not diminish as n increases. For each n there are points close to x = 0 and x = 1 where the error is arbitrarily close to 1.
FIGURE 11.6.2 Dependence of the mean square error Rn on n in Example 1.
Theorem 11.6.1 can be extended to cover self-adjoint boundary value problems having periodic boundary conditions, such as the problem
/ + ÕÓ = 0, (21)
y(—L) — y( L) = 0, y (— L) — Ó( L) = 0 (22)
considered in Example 4 of Section 11.2. The eigenfunctions of the problem (21), (22) are ôï(x) = cos(nnx/L) for n = 0, 1, 2,... and (x) = sin(nnx/L) for n =
1, 2,.... If f is a given square integrable function on — L < x < L, then its expansion
in terms of the eigenfunctions ôï and is of the form
a0 / nn x nn x \
f (x) = 2 + E \an cos~ + bn sin“) ’ (23)
11.6 Series of Orthogonal Functions: Mean Convergence
n = 0, 1, 2,..., (24)
n = 1, 2,.... (25)
This expansion is exactly the Fourier series for f discussed in Sections 10.2 and 10.3.
According to the generalization of Theorem 11.6.1, the series (23) converges in the
mean for any square integrable function f, even though f may not satisfy the conditions of Theorem 10.3.1, which assure pointwise convergence.
> 1. Extend the results of Example 1 by finding the smallest value of n for which Rn < 0.02,