Math Problems Vault

Problem 6.29

Let $f(\vartheta )$ ( $\vartheta$ real) be a real-valued periodic function, $f(\vartheta + 2\pi ) = f(\vartheta )$ . Each of the following equations (or pairs of equations respectively) characterizes a special symmetry property of the graph of the curve $y = f(x)$ :

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(1)

$f(-\vartheta ) = f(\vartheta )$ ;

(2)

$f(-\vartheta ) = -f(\vartheta )$ ;

(3)

$f(\vartheta + \pi ) = -f(\vartheta )$ ;

(4)

$f(-\vartheta ) = f(\vartheta )$ , $f(\vartheta +\pi ) = -f(\vartheta )$ ;

(5)

$f(-\vartheta ) = -f(\vartheta )$ , $f(\vartheta +\pi ) = -f(\vartheta )$ ;

(6)

$f(\vartheta +\pi ) = f(\vartheta )$ .

Show that the existence of such a symmetry property is equivalent to the vanishing of certain (infinitely many) Fourier coefficients.

Problem 6.30

What is the Fourier series of a trigonometric polynomial $f(\vartheta )$ of order $n$ ?

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Problem 6.31

Let $n$ and $\nu$ be positive integers, $0 \leqq \nu \leqq n$ . Then

\frac{1}{2\pi } \int _{-\pi }^{\pi } \left(2 \cos \frac{\vartheta }{2}\right)^n \cos \left(\frac{n}{2} - \nu \right)\vartheta \, d\vartheta = \binom {n}{\nu }.

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Problem 6.32

Let the series of numbers

a_0, a_1, b_1, a_2, b_2, \ldots , a_n, b_n, \ldots

be such that the ``trigonometric series''

a_0 + 2a_1 \cos \vartheta + 2b_1 \sin \vartheta + 2a_2 \cos 2\vartheta + 2b_2 \sin 2\vartheta + \cdots + 2a_n \cos n\vartheta + 2b_n \sin n\vartheta + \cdots

is uniformly convergent for all values of $\vartheta$ and therefore represents a periodic continuous function $f(\vartheta )$ of period $2\pi$ . What is the Fourier series of $f(\vartheta )$ ?

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Problem 6.33

Show that

\frac{1}{2} - \frac{1}{\pi } \sum _{n=1}^{\infty } \frac{\sin 2\pi n x}{n} = \begin{cases} x - [x], & \text{if } x \text{ is not an integer}, \\ \dfrac {1}{2}, & \text{if } x \text{ is an integer}. \end{cases}

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Problem 6.34

Show that

\left|\sin \vartheta \right| = \frac{2}{\pi } - \frac{4}{\pi } \sum _{n=1}^{\infty } \frac{\cos 2n\vartheta }{4n^2 - 1} = \frac{8}{\pi } \sum _{n=1}^{\infty } \frac{\sin ^2 n\vartheta }{4n^2 - 1}.

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Problem 6.35

The constants

\rho _m = \frac{2}{\pi } \int _0^{\pi /2} \frac{\left|\sin m\vartheta \right|}{\sin \vartheta } \, d\vartheta , \qquad m = 1, 2, 3, \ldots

(which are closely related to the so-called Lebesgue constants of the Fourier series) increase steadily with $m$ . They may be represented in the following form:

\rho _m = \frac{16}{\pi ^2} \sum _{n=1}^{\infty } \frac{\dfrac {1}{1} + \dfrac {1}{3} + \dfrac {1}{5} + \cdots + \dfrac {1}{2nm-1}}{4n^2 - 1}.

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Remark.

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Substitute for $\left|\sin m\vartheta \right|$ the series of problem 34 and apply problem 17.

Problem 6.36

Let $f(\vartheta )$ ( $\vartheta$ real) be a real-valued periodic function of period $\pi$ , and in addition an ``even function'', i.e. $f(-\vartheta ) = f(\vartheta )$ . If $f(\vartheta )$ is concave in the interval $0 \leqq \vartheta \leqq \pi$ (see p.65 of Volume I), then its Fourier series has the form

c_0 - c_1 \cos 2\vartheta - c_2 \cos 4\vartheta - \cdots - c_n \cos 2n\vartheta - \cdots ,

where all the coefficients $c_1, c_2, \ldots , c_n, \ldots$ are non-negative.

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Problem 6.37

Let the function $f(\vartheta )$ of problem 36 satisfy in addition the condition $f(0) = 0$ . Further let $\vartheta^{-p} f(\vartheta )$ be bounded, $p > 0$ . Then the sequence

\rho _m = \int _0^{\pi /2} \frac{f(m\vartheta )}{\sin \vartheta } \, d\vartheta , \qquad m = 1, 2, 3, \ldots ,

increases monotonically with $m$ .

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Remark.

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A generalization of problem 35.

Problem 6.38

Let $M_n$ denote the maximum of

\Gamma (n, \vartheta ) = \frac{\left|\sin \vartheta \right|}{1} + \frac{\left|\sin 2\vartheta \right|}{2} + \frac{\left|\sin 3\vartheta \right|}{3} + \cdots + \frac{\left|\sin n\vartheta \right|}{n}

for all values of $\vartheta$ . Then we have the inequalities

\frac{2}{\pi } \sum _{\nu =1}^{n} \frac{1}{\nu } < M_n < \frac{2}{\pi } \sum _{\nu =1}^{n} \frac{1}{\nu } + \frac{2}{\pi }.

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Remark.

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Use problems 34 and 28.