Math Problems Vault

Problem 2.128

If the functions

p_{1}(t), \ p_{2}(t), \ \ldots , \ p_{n}(t), \ \ldots

are continuous on the interval $\left[a,b\right]$ and if they satisfy the conditions

p_{n}(t) \geq 0, \qquad \int _{a}^{b} p_{n}(t) \, dt = 1, \qquad n = 1, 2, 3, \ldots ,

then the terms of the sequence

\int _{a}^{b} p_{1}(t) f(t) \, dt, \qquad \int _{a}^{b} p_{2}(t) f(t) \, dt, \ \ldots , \ \int _{a}^{b} p_{n}(t) f(t) \, dt, \ \ldots

are between the minimum and the maximum of the continuous function $f(t)$ . (Cf. Problems 65, 79, 83 of Part I.)

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Problem 2.129

Let $x$ be a fixed point of the interval $\left[a,b\right]$ considered in Problem 128. In order that

\lim _{n \to \infty } \int _{a}^{b} p_{n}(t) f(t) \, dt = f(x)

holds for all functions $f(t)$ continuous on $\left[a,b\right]$ it is necessary and sufficient that

\lim _{n \to \infty } \left(\int _{a}^{x-\varepsilon } p_{n}(t) \, dt + \int _{x+\varepsilon }^{b} p_{n}(t) \, dt\right) = 0

for all positive values of $\varepsilon$ for which $a < x-\varepsilon < x+\varepsilon < b$ (if $x=a$ or $x=b$ the first or second integral resp. under the limit sign has to be omitted.) (Cf. Problems 66, 80, 84 of Part I.)

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Problem 2.130

We have

\lim _{\varepsilon \to +0} \varepsilon \int _{0}^{\infty } e^{-\varepsilon t} f(t) \, dt = \lim _{t \to \infty } f(t),

provided that the integral on the left and the limit on the right hand side exist.

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Problem 2.131

If the integral

\int _{0}^{\infty } t^{\lambda } f(t) \, dt

converges for $\lambda = \alpha$ and for $\lambda = \beta$ , $\alpha < \beta$ , it converges for $\alpha \leq \lambda \leq \beta$ and it represents a continuous function of $\lambda$ on that interval.

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Problem 2.132

We assume that

p_{1}(x,t), \ p_{2}(x,t), \ \ldots , \ p_{n}(x,t), \ \ldots

are continuous functions of $x$ and $t$ , $a \leq x, t \leq b$ , and that for each $n$

p_{n}(x,t) \geq 0, \qquad \int _{a}^{b} p_{n}(x,t) \, dt = 1 .

Let $f(t)$ denote a continuous function. The functions

f_{n}(x) = \int _{a}^{b} p_{n}(x,t) f(t) \, dt, \qquad n = 1, 2, 3, \ldots

lie between the minimum and the maximum of $f(t)$ on $\left[a,b\right]$ for any $x$ , $a \leq x \leq b$ ; i.e. $\min_{a \leq x \leq b} f(x) \leq f_{n}(x) \leq \max_{a \leq x \leq b} f(x)$ . Furthermore

\lim _{n \to \infty } f_{n}(x) = f(x) \quad \text{for} \quad a < x < b,

provided

\lim _{n \to \infty } \left(\int _{a}^{x-\varepsilon } p_{n}(x,t) \, dt + \int _{x+\varepsilon }^{b} p_{n}(x,t) \, dt\right) = 0

uniformly for $a+\varepsilon \leq x \leq b-\varepsilon$ , $\varepsilon$ fixed and positive. The convergence is uniform on any closed subinterval of $(a,b)$ .

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Problem 2.133

Let $f(x)$ denote a continuous function on $\left[0,1\right]$ . The convergence of

\lim _{n \to \infty } \frac{1}{2} \cdot \frac{3}{2} \cdot \frac{5}{4} \cdots \frac{2n+1}{2n} \int _{0}^{1} f(t) \left[1-(x-t)^{2}\right]^{n} \, dt = f(x)

is uniform for $\varepsilon \leq x \leq 1-\varepsilon$ , $0 < \varepsilon < \frac{1}{2}$ , $\varepsilon$ fixed.

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Problem 2.134

Let $f(x)$ be a continuous periodic function with period $2\pi$ . Then

\lim _{n \to \infty } \frac{1}{2n\pi } \int _{0}^{2\pi } f(t) \left(\frac{\sin n \dfrac {x-t}{2}}{\sin \dfrac {x-t}{2}}\right)^{2} \, dt = f(x),

the convergence is uniform for all $x$ .

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Problem 2.135

Every function defined and continuous on the finite interval $\left[a,b\right]$ can be approximated uniformly on $\left[a,b\right]$ by polynomials to any degree of accuracy. (Weierstrass' approximation theorem.)

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Problem 2.136

Every continuous function that is periodic with period $2\pi$ can be uniformly approximated by trigonometric polynomials (see §2 of Chapter 1, Part VI) to any assigned degree of accuracy. (Weierstrass' approximation theorem.)

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Problem 2.137

Let $f(x)$ denote a function that is properly integrable over $\left[a,b\right]$ ( $\left[0,2\pi \right]$ ). Two polynomials (trigonometric polynomials), $p(x)$ and $P(x)$ , can then be found for any positive $\varepsilon$ so that for $a \leq x \leq b$ ( $0 \leq x \leq 2\pi$ )

p(x) \leq f(x) \leq P(x)

and

\int _{a}^{b} P(x) \, dx - \int _{a}^{b} p(x) \, dx < \varepsilon \qquad \left(\int _{0}^{2\pi } P(x) \, dx - \int _{0}^{2\pi } p(x) \, dx < \varepsilon \right) .

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Problem 2.138

The $n$ -th moment of a function $f(x)$ is given by

\int _{a}^{b} f(t) t^{n} \, dt .

If all the moments of a function that is defined and continuous on the finite interval $\left[a,b\right]$ vanish then the function vanishes identically.

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Problem 2.139

If all the moments

\int _{a}^{b} f(t) t^{n} \, dt, \qquad n = 0, 1, 2, \ldots ,

of a function that is properly integrable over the interval $\left[a,b\right]$ vanish then the function $f(x)$ vanishes at every point of continuity.

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Problem 2.140

If the first $n$ moments vanish,

\int _{a}^{b} f(x) \, dx = \int _{a}^{b} f(x) x \, dx = \int _{a}^{b} f(x) x^{2} \, dx = \cdots = \int _{a}^{b} f(x) x^{n-1} \, dx = 0,

of a function $f(x)$ defined and continuous on the finite or infinite interval $(a,b)$ then the function changes sign (see §2 of Chapter 1, Part V) at least $n$ times in the interval $(a,b)$ unless it is identically $0$ .

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Problem 2.141

The $2n$ -th and $(2n+1)$ -th trigonometric moment (Fourier constants, cf. §4 of Chapter 1, Part VI) of a function with period $2\pi$ are defined as

\int _{0}^{2\pi } \cos nx \, f(x) \, dx \qquad \text{and} \qquad \int _{0}^{2\pi } \sin nx \, f(x) \, dx .

If the first $2n+1$ trigonometric moments of a continuous function $f(x)$ with period $2\pi$ vanish then $f(x)$ changes sign at least $2n+2$ times in any interval of length $> 2\pi$ (see §2 of Chapter 1, Part V) unless $f(x)$ is identically $0$ .

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Problem 2.142

Let the function $\varphi (x)$ be defined and continuous for $x \geq 0$ . Suppose that the integral

J(k) = \int _{0}^{\infty } e^{-kx} \varphi (x) \, dx

converges for $k = k_{0}$ and that it vanishes for a sequence of $k$ 's increasing in arithmetic progression:

J(k_{0}) = J(k_{0}+\alpha ) = J(k_{0}+2\alpha ) = \cdots = J(k_{0}+n\alpha ) = \cdots = 0, \qquad \alpha > 0 .

Then $\varphi (x)$ vanishes identically.

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Problem 2.143

The $\Gamma$ -function

\Gamma (s) = \lim _{n \to \infty } \frac{n^{s} n!}{s(s+1)\cdots (s+n)}

can be written as an integral (Problem 31). Use this fact to prove that $\Gamma (s)$ does not have any zeroes.

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Remark.

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$\Gamma (s+1) = s\Gamma (s)$ (Problem 142).

Problem 2.144

Work out the polynomials $K_{n}(x)$ , $n=0,1,2,\ldots$ for

f(x) = 1, \qquad f(x) = x, \qquad f(x) = x^{2}, \qquad f(x) = e^{x} .

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Problem 2.145

Let $x$ be any point on $\left[0,1\right]$ and

1 = \sum _{v=0}^{n} \binom {n}{v} x^{v}(1-x)^{n-v} = \Sigma ^{\mathrm{I}} + \Sigma ^{\mathrm{II}},

where $\Sigma^{\mathrm{I}}$ refers to the subscripts for which $\left|v-nx\right| \leq n^{3/4}$ and $\Sigma^{\mathrm{II}}$ to those for which $\left|v-nx\right| > n^{3/4}$ , $n \geq 1$ . Then

\Sigma ^{\mathrm{II}} < \frac{1}{4} n^{-\frac{1}{2}} .

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Problem 2.146

Let $f(x)$ be continuous on $\left[0,1\right]$ . The polynomials $K_{n}(x)$ converge uniformly to $f(x)$ on $\left[0,1\right]$ . (New proof of Weierstrass' theorem, Problem 135.)

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