Math Problems Vault

Problem 2.98

We define

g(x) = \sin ^{2}\pi x + \sin ^{2}\pi x \cos ^{2}\pi x + \sin ^{2}\pi x \cos ^{4}\pi x + \cdots + \sin ^{2}\pi x \cos ^{2k}\pi x + \cdots

and

G(x) = \lim _{n \to \infty } g(n! \, x) .

Is $G(x)$ integrable?

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Problem 2.99

Let the function $f(x)$ (cf. also Problem 169 of this Part, Problem 240 of Part VIII) be given by

f(x) = \begin{cases} 0 & \text{if } x \text{ is irrational} \\ \dfrac {1}{q} & \text{if } x \text{ is rational}, \ x = \dfrac {p}{q}, \ (p,q)=1, q \geq 1 . \end{cases}

Show that $f(x)$ is continuous for every irrational $x$ , discontinuous for every rational $x$ and properly integrable over any interval.

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Problem 2.100

The two functions $f(x)$ and $\varphi (x)$ are properly integrable over $\left[a, b\right]$ . Subdivide the interval:

a = x_{0} < x_{1} < x_{2} < \cdots < x_{n-1} < x_{n} = b,

x_{v-1} < y_{v} < x_{v}, \qquad x_{v-1} < \eta _{v} < x_{v}, \qquad v = 1, 2, \ldots , n .

If $\max (x_{v}-x_{v-1}) \to 0$ (the subinterval of maximal length converges to $0$ ) we obtain the relation

\lim _{n \to \infty } \sum _{v=1}^{n} f(y_{v}) \varphi (\eta _{v}) (x_{v}-x_{v-1}) = \int _{a}^{b} f(x) \varphi (x) \, dx .

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Problem 2.101

Suppose that $f(x)$ is properly integrable over $\left[a, b\right]$ and $\varphi (x)$ properly integrable over $\left[a, b+d\right]$ , $d > 0$ . Then

\lim _{\delta \to +0} \int _{a}^{b} f(x) \varphi (x+\delta ) \, dx = \int _{a}^{b} f(x) \varphi (x) \, dx .

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Problem 2.102

Let $f(x)$ denote a properly integrable function on $\left[a, b\right]$ . There exist to every positive number $\varepsilon$ two step-functions, $\psi (x)$ and $\Psi (x)$ , such that for the entire interval $\left[a, b\right]$

\psi (x) \leq f(x) \leq \Psi (x)

and

\int _{a}^{b} \Psi (x) \, dx - \int _{a}^{b} \psi (x) \, dx < \varepsilon .

It is even possible to choose $\psi (x)$ and $\Psi (x)$ so that their points of discontinuity are equidistant.

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Problem 2.103

(continued). If $f(x)$ is of bounded variation $\psi (x)$ and $\Psi (x)$ may be chosen so that the total variation of neither exceeds the total variation of $f(x)$ .

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Problem 2.104

We define

4\left[x\right] - 2\left[2x\right] + 1 = s(x) .

Then we have ( $n$ integer) the limit relation

\lim _{n \to \infty } \int _{0}^{1} f(x) s(nx) \, dx = 0

for any properly integrable function $f(x)$ on $\left[0, 1\right]$ .

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Remark.

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Sketch $s(nx)$ (see Problem 3 of Part VIII).

Problem 2.105

Let $f(x)$ be properly integrable over $\left[a, b\right]$ . Then we can prove that

\lim _{n \to \infty } \int _{a}^{b} f(x) \sin nx \, dx = 0 .

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Problem 2.106

(continued). Yet

\lim _{n \to \infty } \int _{a}^{b} f(x) \left|\sin nx\right| \, dx = \frac{2}{\pi } \int _{a}^{b} f(x) \, dx .

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Problem 2.107

The function $\left(\dfrac {1}{x} - \left[\dfrac {1}{x}\right]\right) x^{\alpha }$ is properly integrable over $\left[0, 1\right]$ for $\alpha \geq 0$ .

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Problem 2.108

If $f(x)$ is properly integrable over $\left[a, b\right]$ the points of continuity of $f(x)$ are everywhere dense on this interval.

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Problem 2.109

The function $f(x)$ is properly integrable over $\left[a, b\right]$ . The equation

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

is satisfied if and only if $f(\xi ) = 0$ at every point of continuity $\xi$ of $f(x)$ , $a < \xi < b$ .

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Problem 2.110

Assume that the function $y = f(x)$ is properly integrable over $\left[a, b\right]$ , that $m \leq f(x) \leq M$ on this interval and that the function $\varphi (y)$ is continuous on $\left[m, M\right]$ . Then also $\varphi \left[f(x)\right]$ is properly integrable over $\left[a, b\right]$ .

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Problem 2.111

If $f(x)$ and $\varphi (y)$ are properly integrable $\varphi \left[f(x)\right]$ need not be properly integrable (see Problems 98, 99).

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