Math Problems Vault

Problem 6.50

Let

g(\vartheta ) = \lambda _0 + \lambda _1 \cos \vartheta + \mu _1 \sin \vartheta + \lambda _2 \cos 2\vartheta + \mu _2 \sin 2\vartheta + \cdots + \lambda _n \cos n\vartheta + \mu _n \sin n\vartheta

( $\vartheta$ real) be a real non-negative trigonometric polynomial of $n$ th order, and let the constant term in $g(\vartheta )$ be equal to $1$ , i.e.

\lambda _0 = \frac{1}{2\pi } \int _0^{2\pi } g(\vartheta ) \, d\vartheta = 1.

Then

g(\vartheta ) \leqq n+1.

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Problem 6.51

With the notation of problem 50, we have

\sqrt{\lambda _n^2 + \mu _n^2} \leqq 1.

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Problem 6.52

With the notation of problem 50, we have

\sqrt{\lambda _1^2 + \mu _1^2} \leqq 2\cos \frac{\pi }{n+2}.

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Problem 6.53

Let $h(z) = c(1+z_1 z)(1+z_2 z) \cdots (1+z_n z)$ , $\left|z_\nu \right| \leqq 1$ , $\nu = 1, 2, \ldots , n$ , $c$ real, $c > 0$ . Then, setting $g(\vartheta ) = \left|h(e^{i\vartheta })\right|^2$ , we have

\frac{1}{2\pi } \int _0^{2\pi } \log g(\vartheta ) \, d\vartheta = \log c^2.

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Problem 6.54

Let $g(\vartheta ) = \left|x_0 + x_1 z + x_2 z^2 + \cdots + x_n z^n\right|^2$ , $z = e^{i\vartheta }$ , where $x_0, x_1, \ldots , x_n$ range over all complex numbers such that

\frac{1}{2\pi }\int _0^{2\pi } \log g(\vartheta ) \, d\vartheta = 0

(that is, with the notation of problem 53, $c = 1$ ). Determine the maximum of $g(\vartheta_0)$ , $\vartheta_0$ fixed.

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Problem 6.55

With the notation of problems 53, 54, determine the maximum of $\lambda_0$ .

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Problem 6.56

With the notation of problems 53, 54, determine the maximum of $\sqrt{\lambda_\nu^2 + \mu_\nu^2}$ , $\nu = 1, 2, \ldots , n$ .

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Problem 6.57

A real trigonometric polynomial without a constant term

g(\vartheta ) = \lambda _1 \cos \vartheta + \mu _1 \sin \vartheta + \lambda _2 \cos 2\vartheta + \mu _2 \sin 2\vartheta + \cdots + \lambda _n \cos n\vartheta + \mu _n \sin n\vartheta

cannot have the same sign for all real $\vartheta$ unless it is identically zero. This theorem is to be proved without the use of integral calculus.

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Problem 6.58

Let $-m$ and $M$ be the minimum and maximum, respectively, of a real trigonometric polynomial of $n$ th order of the form

g(\vartheta ) = \lambda _1 \cos \vartheta + \mu _1 \sin \vartheta + \lambda _2 \cos 2\vartheta + \mu _2 \sin 2\vartheta + \cdots + \lambda _n \cos n\vartheta + \mu _n \sin n\vartheta ,

( $\vartheta$ real) $m \geqq 0$ , $M \geqq 0$ (problem 57). Then we have

M \leqq nm, \qquad m \leqq nM.

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Problem 6.59

The first mean-value theorem of integral calculus may be strengthened for real trigonometric polynomials as follows: Let $g(\vartheta )$ be a real trigonometric polynomial of $n$ th order and let $m$ be the minimum and $M$ the maximum of $g(\vartheta )$ ( $\vartheta$ real). Then we have

m + \frac{M-m}{n+1} \leqq \frac{1}{2\pi } \int _0^{2\pi } g(\vartheta ) \, d\vartheta \leqq M - \frac{M-m}{n+1}.

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Problem 6.60

Let $-m$ be the minimum and $M$ the maximum of a real trigonometric polynomial of $n$ th order

g(\vartheta ) = \lambda _0 + \lambda _1 \cos \vartheta + \mu _1 \sin \vartheta + \lambda _2 \cos 2\vartheta + \mu _2 \sin 2\vartheta + \cdots + \lambda _n \cos n\vartheta + \mu _n \sin n\vartheta ,

( $\vartheta$ real). Then either $m$ or $M$ is greater than $\sqrt{\lambda_n^2 + \mu_n^2}$ . We have $m = M = \sqrt{\lambda_n^2 + \mu_n^2}$ only if $g(\vartheta ) = c\cos n(\vartheta - \vartheta_0)$ .

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Problem 6.61

Compound $n$ simple harmonic motions with periods in the ratio

1 : \frac{1}{2} : \frac{1}{3} : \cdots : \frac{1}{n}

and with arbitrary phases. The greatest displacement of the resulting motion is then equal to at least the arithmetic mean of the amplitudes of the separate simple harmonic motions.

With the notation of problem 58, we are dealing with the inequality:

M \geqq \frac{\sqrt{\lambda _1^2+\mu _1^2} + \sqrt{\lambda _2^2+\mu _2^2} + \cdots + \sqrt{\lambda _n^2+\mu _n^2}}{n}.

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Remark.

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Strengthening of problem 50.