Math Problems Vault

Problem 7.38 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.3 "Generation of Positive Quadratic Forms"

Let the real numbers $a_{0}, a_{1}, a_{2}, \ldots , a_{2n}$ have the following property: If $f(x)$ denotes an arbitrary polynomial, with real coefficients of degree at most $2n$ , that does not vanish identically and does not take a negative value for any real value of $x$ , then also

a_{0} f(x) + \frac{a_{1}}{1!} f'(x) + \frac{a_{2}}{2!} f''(x) + \cdots + \frac{a_{2n}}{(2n)!} f^{(2n)}(x) \geq 0 \qquad (\text{or} > 0)

for all real values of $x$ .

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

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} a_{\lambda +\mu } x_{\lambda } x_{\mu }

is positive (or positive definite, respectively).

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Problem 7.39 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.3 "Generation of Positive Quadratic Forms"

Let the numbers $a_{0}, a_{1}, \ldots , a_{n}$ , $n \geq 1$ have the following property: If $f(x)$ is an arbitrary polynomial with real coefficients of degree at most $n$ , that does not vanish identically and is non-negative for non-negative values of $x$ , then also

a_{0} f(x) + \frac{a_{1}}{1!} f'(x) + \frac{a_{2}}{2!} f''(x) + \cdots + \frac{a_{n}}{n!} f^{(n)}(x) \geq 0 \qquad (\text{or} > 0)

for all non-negative values of $x$ ( $x$ real).

Show that for this to be true it is necessary and sufficient that the two quadratic forms

\sum _{\lambda =0}^{[n/2]} \sum _{\mu =0}^{[n/2]} a_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{[(n-1)/2]} \sum _{\mu =0}^{[(n-1)/2]} a_{\lambda +\mu +1} x_{\lambda } x_{\mu }

are positive (or positive definite, respectively).

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Problem 7.40 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.3 "Generation of Positive Quadratic Forms"

Let the numbers $a_{0}, a_{1}, a_{2}, \ldots , a_{n}$ , $n \geq 1$ have the following property: If $f(x)$ denotes a polynomial with real coefficients of degree $\leq n$ that does not vanish identically and is non-negative in the interval $-1 \leq x \leq 1$ , then also

a_{0} f(x) + \frac{a_{1}}{1!} f'(x) + \frac{a_{2}}{2!} f''(x) + \cdots + \frac{a_{n}}{n!} f^{(n)}(x) \geq 0

for $-1 \leq x \leq 1$ . Show that the only set of numbers $a_{0}, a_{1}, a_{2}, \ldots , a_{n}$ of this kind is determined by the conditions

a_{0} \geq 0, \qquad a_{1} = a_{2} = \cdots = a_{n} = 0.

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Problem 7.41 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.3 "Generation of Positive Quadratic Forms"

Let the two quadratic forms

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} a_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} b_{\lambda +\mu } x_{\lambda } x_{\mu }

be positive. If we set

c_{\nu } = a_{0} b_{\nu } + \binom {\nu }{1} a_{1} b_{\nu -1} + \binom {\nu }{2} a_{2} b_{\nu -2} + \cdots + a_{\nu } b_{0},

then the quadratic form

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\lambda +\mu } x_{\lambda } x_{\mu }

is also positive, and it is in fact definite if at least one of the two preceding forms is definite and the other not identically vanishing.

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Problem 7.42 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.3 "Generation of Positive Quadratic Forms"

Let

c_{\nu } = a_{0} b_{\nu } + \binom {\nu }{1} a_{1} b_{\nu -1} + \binom {\nu }{2} a_{2} b_{\nu -2} + \cdots + a_{\nu } b_{0}.

From the positiveness of the four quadratic forms

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} a_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m-1} \sum _{\mu =0}^{m-1} a_{\lambda +\mu +1} x_{\lambda } x_{\mu },

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} b_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m-1} \sum _{\mu =0}^{m-1} b_{\lambda +\mu +1} x_{\lambda } x_{\mu }

follows the positiveness of the following two forms:

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} c_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m-1} \sum _{\mu =0}^{m-1} c_{\lambda +\mu +1} x_{\lambda } x_{\mu }.

Furthermore, from the positiveness of the four quadratic forms

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} a_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} a_{\lambda +\mu +1} x_{\lambda } x_{\mu },

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} b_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} b_{\lambda +\mu +1} x_{\lambda } x_{\mu }

follows the positiveness of the following two forms:

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} c_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} c_{\lambda +\mu +1} x_{\lambda } x_{\mu }.

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Problem 7.43 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.3 "Generation of Positive Quadratic Forms"

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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Moment problems and positivity of Hankel and Toeplitz forms

References