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? 1. Extend the results of Example 1 by finding the smallest value of n for which Rn < 0.02,
where Rn is given by Eq. (20).
? 2. Let f (x) = x for 0 < x < 1 and let 0m(x) = \f2sin mnx.
(a) Find the coefficients bm in the expansion of f (x) in terms of 01 (x), 02(x), ....
(b) Calculate the mean square error Rn for several values of n and plot the results.
(c) Find the smallest value of n for which Rn < 0.01.
? 3. Follow the instructions for Problem 2 using f (x) = x(1 — x) for 0 < x < 1.
4. In this problem we show that pointwise convergence of a sequence Sn (x) does not imply
mean convergence, and conversely.
(a) Let Sn (x) = n*J~xe~nx /2, 0 < x < 1. Show that Sn (x) ^ 0 as n for each x
in 0 < x < 1. Show also that
Rn = j'[0 — Sn(x)]2 dx = 2(1 — e—n),
and hence that Rn as n ^(Xi. Thus pointwise convergence does not imply mean
(b) Let Sn (x) = xn for 0 < x < 1 and let f (x) = 0 for 0 < x < 1. Show that
Rn = jf[ f(x) — Sn(x)]2 dx = zn+T•
and hence Sn (x) converges to f (x) in the mean. Also show that Sn (x) does not converge to f(x) pointwise throughout 0 < x < 1. Thus mean convergence does not imply pointwise convergence.
5. Suppose that the functions 01,...,0n satisfy the orthonormality relation (1), and that a given function f is to be approximated by Sn (x) = c101(x) + ••• + cn0n (x), where the coefficients ct are not necessarily those of Eq. (9). Show that the mean square error Rn given by Eq. (6) may be written in the form
C1 n n
Rn = j r(x) f2 (x) dx —J2 a2 + J2 c — a!)2,
J0 i=1 i=1
where the at are the Fourier coefficients given by Eq. (9). Show that Rn is minimized if Cj = a. for each i.
6. In this problem we show by examples that the (Riemann) integrability of f and f2 are independent.
1 fL nn x
L. l-L f (x) cos —j—
1 fl nn x
L. L f (x) sin —j—
bn = — I f (x) sin —— dx,
Chapter 11. Boundary Value Problems and Sturm Liouville Theory
m t t ff \ \x 1/2, 0 < x < 1,
(a) Let f(x) = jQ, x = 07
Show that J“ f (x) dx exists as an improper integral, but J“ f2 (x) dx does not.
(b) Let f (x) = ) 1’ x rational,
[ —1, x irrational.
Show that f2 (x) dx exists, but f (x) dx does not.
7. Suppose that it is desired to construct a set of polynomials f0(x), fi(x), f2(x), fk(x), ..., where fk(x) is of degree k, that are orthonormal on the interval 0 < x < 1. That is, the set of polynomials must satisfy
(f,’ k) = \ f ,(x) fk(x) dx = j.
(a) Find f0(x) by choosing the polynomial of degree zero such that (f0, f0) = 1.
(b) Find f1(x) by determining the polynomial of degree one such that (f0, Zj) = 0 and (fj, fj) = 1.
(c) Find f2(x).
(d) The normalization condition (fk, fk) = 1 is somewhat awkward to apply. Let g0(x), g1(x), ..., gk (x),... be the sequence of polynomials that are orthogonal on 0 < x < 1 and that are normalized by the condition gk(1) = 1. Find g0(x), g1(x), and g2(x) and compare them with f0(x), f1(x),and f2(x).
8. Suppose that it is desired to construct a set of polynomials P0(x), P1 (x), ..., Pk(x),..., where p (x) is of degree k, that are orthogonal on the interval — 1 < x < 1; see Problem 7. Suppose further that Pk(x) is normalized by the condition p(1) = 1. Find P0(x), P1(x), P2(x), and P3(x). Note that these are the first four Legendre polynomials (see Problem 24 of Section 5.3).
9. This problem develops some further results associated with mean convergence. Let Rn(a1, ..., an), Sn(x), and at be defined by Eqs. (6), (2), and (9), respectively.
(a) Show that
r 1 n
Rn = r (x) f 2(x) dx -T al
Hint: Substitute for Sn(x) in Eq. (6) and integrate, using the orthogonality relation (1).
(b) Show that fa2 < Ir(x) f2(x) dx. This result is known as Bessel’s inequality.
(c) Show that ai2 converges.
f 1 ^
(d) Show that lim R = I r(x) f2(x) dx — a2.
n^TO n Jo i=1
(e) Show that ^ at $>j (x) converges to f (x) in the mean if and only if
i=1 i i
f 1 TO
/ r(x) f2(x) dx =V a2.
0 i=1 i
This result is known as Parseval’s equation.
In Problems 10 through 12 let p, $2,..., 0n, ... be the normalized eigenfunctions of the
Sturm-Liouville problem (11), (12).
10. Show that if an is the nth Fourier coefficient of a square integrable function f, then lim an = 0.
Hint: Use Bessel’s inequality, Problem 9(b).
11.6 Series of Orthogonal Functions: Mean Convergence
11. Show that the series
01(x) + 02(x) +--------+0n (x) +-----
cannot be the eigenfunction series for any square integrable function.
Hint: See Problem 10.
12. Show that the series
0(x) + -TT + + •••
is not the eigenfunction series for any square integrable function.
Hint: Use Bessel’s inequality, Problem 9(b).
13. Show that Parseval’s equation in Problem 9(e) is obtained formally by squaring the series (10) corresponding to f, multiplying by the weight function r, and integrating term by term.
The following books were mentioned in the text in connection with certain theorems about Sturm-Liouville problems: