Math Problems Vault

Problem 4.1.1

Show that the trigonometric system

\frac{1}{\sqrt{2 \pi }}, \quad \frac{\cos (n x)}{\sqrt{\pi }}, \frac{\sin (m x)}{\sqrt{\pi }}, \quad n, m=1,2, \ldots

is an orthonormal sequence in $L^{2}([-\pi , \pi ], m)$ .

?

Problem 4.1.2

(a)(b)(c)(d)

In this exercise you will actually compute a classical (i.e., with respect to the orthonormal sequence of Exercise 1) Fourier series, and investigate its convergence properties. The function given is a basic one; in the next exercise you are asked to do the same procedure with another very basic function. You are being asked to do these by hand, and you can no doubt appreciate that the computations get quite laborious once we depart from even the most basic functions. There are tricks for doing these computations; the interested reader can learn more about such techniques in a text devoted to classical Fourier series.

?

(a)

f(x)= \begin{cases} 1 & \text{ if }-\pi \leq x<0 \\ 0 & \text{ if } 0 \leq x<\pi \end{cases}

Show that its Fourier series is

\frac{1}{2}+\frac{1}{\pi } \sum _{n=1}^{\infty } \frac{\left((-1)^{n}-1\right)}{n} \sin (n x)

(b)

Explain why this series converges in mean to $f$ .

(c)

What can you say about the pointwise and uniform convergence of this series?

(d)

Why are the coefficients of the cosine terms all zero?

Problem 4.1.3

(a)(b)(c)(d)

In this exercise you will compute another Fourier series and investigate its convergence properties.

?

(a)

Let $f(x)=x^{2}$ . Show that its classical Fourier series is

\frac{\pi ^{2}}{3}+4 \sum _{n=1}^{\infty } \frac{(-1)^{n}}{n^{2}} \cos (n x)

(b)

Explain why this series converges in mean to $f$ .

(c)

What can you say about the pointwise and uniform convergence of this series?

(d)

Why are the coefficients of the sine terms all zero?

Problem 4.1.4

For a sequence $\left\{ f_{n}\right\}_{n=1}^{\infty }$ in $L^{2}([-\pi , \pi ], m)$ , we have seen three ways for $\left\{ f_{n}\right\}_{n=1}^{\infty }$ to converge:

"pointwise,"
"uniformly,"
"in mean."

The point of this exercise is to understand the relation between these three types of convergence. For the counterexamples asked for below, use whatever finite interval $[a, b]$ you find convenient. Please make an effort to supply "easy" examples.

?

(a)

Prove that uniform convergence implies pointwise convergence.

(b)

Give an example to show that pointwise convergence does not imply uniform convergence.

(c)

Prove that uniform convergence implies convergence in mean.

(d)

Give an example to show that pointwise convergence does not imply convergence in mean.

(e)

Give an example to show that convergence in mean does not imply pointwise convergence. (Note that the same example shows that convergence in mean does not imply uniform convergence.)

Problem 4.1.5

Apply the Gram-Schmidt process to the functions $1, x, x^{2}, x^{3}, \ldots$ to obtain formulas for the first three Legendre polynomials. Then verify that they are indeed given by the formula

\sqrt{\frac{2 n+1}{2}} \frac{1}{2^{n} n !} \frac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n}, \quad n=1,2,3

?

Problem 4.1.6

Prove that the Haar family is an orthonormal family in the Hilbert space $L^{2}([0,1])$ .

?

Problem 4.1.7

(a)(b)

?

(a)

Show that the sequence

\frac{e^{i n x}}{\sqrt{2 \pi }}, \quad n=0, \pm 1 \pm 2, \ldots

is a complete orthonormal sequence in $L^{2}([-\pi , \pi ])$ .

(b)

Show that the sequence

\sqrt{\frac{2}{\pi }} \cos (n x), \quad n=1,2,3, \ldots

is a complete orthonormal sequence in $L^{2}([0, \pi ])$ . (Observe that $\sqrt{\frac{2}{\pi }} \cos (n x)$ can be replaced by $\sqrt{\frac{2}{\pi }} \sin (n x)$ .)