Math Problems Vault

Problem 7.43

Let $\vartheta$ be real and let the complex numbers

c_{-n}, c_{-n+1}, \ldots , c_{-1}, c_{0}, c_{1}, \ldots , c_{n-1}, c_{n}, \qquad c_{-\nu } = \bar{c}_{\nu }, \quad \nu = 0, 1, 2, \ldots , n

have the following property: If

g(\vartheta ) = \alpha _{0} + 2(\alpha _{1} \cos \vartheta + \beta _{1} \sin \vartheta + \alpha _{2} \cos 2\vartheta + \beta _{2} \sin 2\vartheta + \cdots + \alpha _{n} \cos n\vartheta + \beta _{n} \sin n\vartheta )

= \sum _{\nu =-n}^{n} \gamma _{\nu } e^{-i\nu \vartheta }, \qquad \gamma _{\nu } = \bar{\gamma }_{-\nu } = \alpha _{\nu } + i\beta _{\nu }, \quad \nu = 0, 1, 2, \ldots , n; \quad \beta _{0} = 0,

denotes an arbitrary trigonometric polynomial of degree at most $n$ , that does not vanish identically and does not have a negative value for any value of $\vartheta$ , let

\sum _{\nu =-n}^{n} c_{\nu } \gamma _{\nu } e^{-i\nu \vartheta } \geq 0 \qquad (\text{or} > 0).

Show that for this to be true, it is necessary and sufficient that the Hermitian form

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\mu -\lambda } x_{\lambda } \bar{x}_{\mu }

is positive (or positive definite, respectively).

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Problem 7.43.1

Let $G(z)$ be an entire function that takes real values for real $z$ , with $G(0) = 1$ , and

-\frac{G'(z)}{G(z)} = s_{1} + s_{2} z + s_{3} z^{2} + \cdots + s_{m} z^{m-1} + \cdots .

If $G(z)$ has infinitely many zeros, all real, and is of the form given in Problem 165 of Part V, then, for $n = 1, 2, 3, \ldots$ ,

\begin{vmatrix} s_{2} & s_{3} & s_{4} & \ldots & s_{n+1} \\ s_{3} & s_{4} & s_{5} & \ldots & s_{n+2} \\ s_{4} & s_{5} & s_{6} & \ldots & s_{n+3} \\ \multicolumn {5}{c}{\dotfill } \\ s_{n+1} & s_{n+2} & s_{n+3} & \ldots & s_{2n} \end{vmatrix} > 0.

(If $G(z)$ of the form in Problem 165 of Part V has only a finite number of zeros, then only a finite number of these recurrent determinants can be $\neq 0$ [Problem 27].)

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Problem 7.43.2

Let $G(z)$ be an entire function of the form given in Problem 165 of Part V, but not of the form $e^{\beta z}$ , and

\frac{1}{G(z)} = c_{0} + \frac{c_{1}}{1!} z + \frac{c_{2}}{2!} z^{2} + \frac{c_{3}}{3!} z^{3} + \cdots .

Then, for $n = 0, 1, 2, \ldots$ ,

\begin{vmatrix} c_{0} & c_{1} & c_{2} & \ldots & c_{n} \\ c_{1} & c_{2} & c_{3} & \ldots & c_{n+1} \\ c_{2} & c_{3} & c_{4} & \ldots & c_{n+2} \\ \multicolumn {5}{c}{\dotfill } \\ c_{n} & c_{n+1} & c_{n+2} & \ldots & c_{2n} \end{vmatrix} > 0.

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