33.8. EXERCISES 665

28. Recall the normalized Legendre polynomials

qn (x) =√

2n+1√2

pn (x)

which have the property that

∫ 1

−1q j (x)qk (x)dx = δ jk =

{1 if j = k0 if j ̸= k

If f is a Riemann integrable function, show that

limn→∞

∫ 1

−1f (x)qn (x)dx = 0

29. Show that if f is any continous function on [−1,1] , then the Fourier series in termsof Legendre polynomials converges to f in the mean square sense. This means thatfor Sn f (x)≡ ∑

nk=0 ck pk (x) ,ck =

∫ 1−1 qk (x) f (x)dx, it follows that

limn→∞

∫ 1

−1| f (x)−Sn f (x)|2 dx = 0

30. It can be shown that there are no continuous, nonzero solutions to Legendre’s equa-tion ((

1− x2)y′)′+λy = 0

defined on [−1,1] unless λ = n(n+1) for n an integer. Use the above problem toshow this.

31. One of the applications of Fourier series is to obtain solutions to linear differentialequations which have a periodic right side. This is done by expanding the right sidewhich is a forcing function in a Fourier series, solving the simple equation whichcorresponds to each term and then adding these solutions to obtain what is hopedto be a representation of the solution. Find a particular solution for each of thefollowing. Let

y′′+3y = f (t) ,

where f (t) is the step function which is periodic of period 2 and equals −1 on[−1,0) and 1 on (0,1]. Here are the steps. First find a Fourier series for f . Say∑

∞n=1 bn sin(nπx) . Then let yn be the solution to

y′′n +3yn = sin(nπt)

and then hopefully, on neglecting mathematical issues, the solution to the originalproblem is

y(t) =∞

∑n=1

bnyn (t)

32. Explain why the above procedure should give a particular solution if mathematicalissues related to interchange of limit operations are ignored.