Math Problems Vault

Problem 6.103

Let $P(x)$ be an arbitrary polynomial of $n$ th degree with real coefficients satisfying

\int _{-1}^{1} [P(x)]^2\, dx = 1.

Then for $-1 \leqq x \leqq 1$ we have

\left|P(x)\right| \leqq \frac{n+1}{\sqrt{2}}.

The equality sign here holds if and only if either

P(x) = \pm \frac{\sqrt{2}}{n+1}S_n(x) \quad \text{(problem 88)} \quad \text{and} \quad x=1

or

P(x) = \pm \frac{\sqrt{2}}{n+1}S_n(-x) \quad \text{and} \quad x=-1

( $n>0$ ).

?

Remark.

?

The integral of $[P(x)]^2$ is a quadratic form in $n+1$ variables. Choose these variables in such a manner that the quadratic form becomes a sum of $n+1$ squares.

Problem 6.104

Let $P(x)$ be an arbitrary polynomial of $n$ th degree with real coefficients satisfying

\int _{-1}^{1} (1-x)[P(x)]^2\, dx = 1.

Then we have

\left|P(1)\right| \leqq \frac{1}{\sqrt{2}}\binom {n+2}{2}, \qquad \left|P(-1)\right| \leqq \frac{1}{\sqrt{2}}\sqrt{\binom {n+2}{2}}.

These bounds cannot be replaced by any smaller bounds.

?

Problem 6.105

Let $\alpha$ and $\beta$ be constants, $\alpha >-1$ , $\beta >-1$ , and $P(x)$ an arbitrary polynomial of $n$ th degree with real coefficients satisfying

\int _{-1}^{1} (1-x)^\alpha (1+x)^\beta [P(x)]^2\, dx = 1.

Determine the maximum of $\left|P(1)\right|$ and $\left|P(-1)\right|$ , as $P(x)$ ranges through the set of all polynomials of the specified kind. How do these maximum values behave for large values of $n$ ?

?

Problem 6.106

Let $\alpha$ be a constant, $\alpha >-1$ . Determine the maximum of $\left|P(0)\right|$ for all polynomials $P(x)$ of $n$ th degree which satisfy the condition

\int _0^{\infty } e^{-x}x^\alpha [P(x)]^2\, dx = 1.

How does this maximum behave for large values of $n$ ?

?

Problem 6.107

Determine the maximum of $\left|P(0)\right|$ for all polynomials $P(x)$ of $n$ th degree, which satisfy the condition

\int _{-\infty }^{\infty } e^{-\frac{x^2}{2}}[P(x)]^2\, dx = 1.

How does this maximum behave for large values of $n$ ?

?

Problem 6.108

Let $P(x)$ be a polynomial of $n$ th degree which assumes only non-negative values in the interval $-1 \leqq x \leqq 1$ , and which satisfies

\int _{-1}^{1} P(x)\, dx = 1.

Then we have

P(1) \leqq \begin{cases} \dfrac {q(q+1)}{2} & \text{for odd } n; \quad n=2q-1, \\ ]6pt] \dfrac {(q+1)^2}{2} & \text{for even } n, \quad n=2q. \end{cases}

The same estimate holds for $P(-1)$ . These bounds cannot be replaced by any smaller bounds.

?

Problem 6.109

The first mean-value theorem of integral calculus may be strengthened for polynomials of $n$ th degree as follows: Let $P(x)$ be a polynomial of $n$ th degree and $m$ the minimum, $M$ the maximum of $P(x)$ in the interval $a \leqq x \leqq b$ . Then we have

m + \frac{M-m}{\alpha _n} \leqq \frac{1}{b-a}\int _a^b P(x)\, dx \leqq M - \frac{M-m}{\alpha _n},

where $\alpha_n = q(q+1)$ for odd $n$ , $n=2q-1$ , and $\alpha_n = (q+1)^2$ for even $n$ , $n=2q$ .

?

Problem 6.110

Let $\alpha$ and $\beta$ be constants, $\alpha >-1$ , $\beta >-1$ , and $P(x)$ a polynomial of $n$ th degree which assumes only non-negative values in the interval $-1 \leqq x \leqq 1$ and which satisfies

\int _{-1}^{1} (1-x)^\alpha (1-x)^\beta P(x)\, dx = 1.

Then we have

P(1) \leqq \begin{cases} \dfrac {1}{2^{\alpha +\beta +1}}\dfrac {\Gamma (q+\alpha +1)\Gamma (q+\alpha +\beta +2)}{\Gamma (\alpha +1)\Gamma (\alpha +2)\Gamma (q)\Gamma (q+\beta +1)} & \text{for odd } n, \quad n=2q-1, \\ ]8pt] \dfrac {1}{2^{\alpha +\beta +1}}\dfrac {\Gamma (q+\alpha +2)\Gamma (q+\alpha +\beta +2)}{\Gamma (\alpha +1)\Gamma (\alpha +2)\Gamma (q+1)\Gamma (q+\beta +1)} & \text{for even } n, \quad n=2q. \end{cases} \[ These bounds cannot be replaced by any smaller bounds. The corresponding bounds for $P(-1)$ are obtained by interchanging $\alpha $ and $\beta $. \]

?

Problem 6.111

Let $\alpha$ be a constant, $\alpha >-1$ , and $P(x)$ a polynomial of $n$ th degree which assumes only non-negative values for non-negative values of $x$ , and which satisfies

\int _0^{\infty } e^{-x}x^\alpha P(x)\, dx = 1.

Then we have

P(0) \leqq \frac{\Gamma (p+\alpha +2)}{\Gamma (\alpha +1)\Gamma (\alpha +2)\Gamma (p+1)}, \qquad p = \left[\frac{n}{2}\right].

This bound cannot be replaced by any smaller bound.

?

Problem 6.112

Let $P(x)$ be a polynomial of $n$ th degree, which is non-negative for non-negative values of $x$ , and which satisfies

\int _0^{\infty } e^{-x}P(x)\, dx = 1.

Then we have

P(0) \leqq \left[\frac{n}{2}\right] + 1.

?

Problem 6.113

(Continuation of problem 112.) Let $\xi$ be an arbitrary non-negative number. Then we have

e^{-\xi }P(\xi ) \leqq \left[\frac{n}{2}\right]+1.

?