All textbooks/Selected Problems in Real Analysis

Chapter IX

Sequences of Measurable Functions

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IX.1 · Convergence in Measure and Almost Everywhere

IX.2 · Convergence in the Mean. The Law of Large Numbers

IX.3 · Rademacher Functions. The Khinchin Inequality

IX.4 · Fourier Series and Transform

Section IX.1Convergence in Measure and Almost Everywhere[16/81%]

Problem IX.1.1

(a)(b)(c)(d)(e)(f)(g)

Do the functional sequences $\left\{ f_{n}\right\}$ indicated below converge in measure and almost everywhere on the intervals $\Delta$ ? Do they have summable majorants?

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(a)

$f_{n}(x)=\frac{n x}{1+n^{2} x^{2}}$ , $\Delta =(0,1) ;$

(b)

$f_{n}(x)=\frac{n x}{1+n^{2} x^{2}}$ , $\Delta =(1,+\infty ) ;$

(c)

$f_{n}(x)=n x e^{-n x^{2}}$ , $\Delta =(0,1) ;$

(d)

$f_{n}(x)=n^{2} x e^{-n x^{2}}$ , $\Delta =(1,+\infty ) ;$

(e)

$f_{n}(x)=\frac{n^{2} x}{n^{4}+x^{2}}$ , $\Delta =(1,+\infty ) ;$

(f)

$f_{n}(x)=\frac{1}{\ln (n+1)} \cdot \frac{n x}{1+n^{2} x^{4}}$ , $\Delta =(0,1) ;$

(g)

$f_{n}(x)=\left|x-\frac{1}{n}\right|^{-1 / 2}$ , $\Delta =(0,1)$ .

Problem IX.1.2

(a)(b)

Let $a \in \mathbb {R},\left\{ a\right\} =a-\left[a\right]$ be the fractional part of the number $a$ , and let $f_{n}$ be the characteristic function of the interval with endpoints $\left\{ \ln n\right\}$ , $\left\{ \ln (n+1)\right\} (n \in \mathbb {N})$ (if $\left\{ \ln n\right\} >\left\{ \ln (n+1)\right\}$ , we take $f_{n}$ to be identically zero). Prove that

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(a)

$\varlimsup_{n \rightarrow \infty } f_{n}(x)=1$ for any $x \in (0,1)$ ;

(b)

$f_{\left[C^{k}\right]}(x) \underset {k \rightarrow \infty }{\longrightarrow } 0$ almost everywhere on $(0,1)$ for any $C>1$ .

Compare this result with the statement of Problem II.1.28.

Problem IX.1.3

Construct a sequence $\left\{ g_{n}\right\}$ from $\mathscr {L}^{0}(0,1)$ such that $g_{n} \rightarrow 0$ in measure, $\varlimsup_{n \rightarrow \infty } g_{n}(x)=+\infty$ almost everywhere on $(0,1)$ , and $\lim_{n \rightarrow \infty } g_{n}(x)=-\infty$ almost everywhere on $(0,1)$ .

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Problem IX.1.4

Let the functions $f_{n}(n \in \mathbb {N})$ be monotone on the interval $(0,1)$ and $f_{n} \rightarrow f$ in measure. Prove that the function $f$ coincides almost everywhere with a monotone function $f_{0}$ and $f_{n}(x) \rightarrow f_{0}(x)$ at the points of continuity of $f_{0}$ .

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Problem IX.1.5

Let $f_{n}(x)=\left|\sin \left(A_{n} x+\varphi_{n}\right)\right|^{p_{n}}$ , where $x \in (0,2 \pi )$ , and $\left\{ A_{n}\right\} ,\left\{ \varphi_{n}\right\}$ , $\left\{ p_{n}\right\}$ are real numerical sequences satisfying the conditions $\lim \left|A_{n}\right|>0$ and $p_{n} \rightarrow +\infty$ . Prove that $f_{n} \rightarrow 0$ in measure on $(0,2 \pi )$ . Must $f_{n} \rightarrow 0$ almost everywhere on $(0,2 \pi )$ ? Consider the example: $f_{n}(x)=\left|\cos \sqrt{m}\left(x-\frac{k \pi }{m}\right)\right|^{m}$ , where $m=\left[\sqrt{n}\right], k=n-m^{2}$ .

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Problem IX.1.6

Let $\left\{ A_{n}\right\} ,\left\{ \varphi_{n}\right\}$ be real numerical sequences, $A_{n} \rightarrow +\infty$ . Prove that almost everywhere on $\mathbb {R}$

\varlimsup _{n \rightarrow \infty } \cos \left(A_{n} x+\varphi _{n}\right)=1, \quad \lim _{n \rightarrow \infty } \cos \left(A_{n} x+\varphi _{n}\right)=-1 .

Compare this statement with Problem IV.6.1.

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Problem IX.1.7

Prove that $n^{1 / 3} \int_{0}^{\infty } e^{-x} \sin^{n}\left(n x+\varphi_{n}\right) d x \rightarrow 0$ for any real sequence $\left\{ \varphi_{n}\right\}$ .

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Problem IX.1.8

Let $\left\{ f_{n}\right\} \subset \mathscr {L}^{0}(\mathbb {R}), a \in \mathbb {R}$ . Prove that $\varlimsup_{n \rightarrow \infty } f_{n}(x) \leqslant a$ almost everywhere on $\mathbb {R}$ , if $\sum_{n \geqslant 1} \lambda \left\{ x \in \mathbb {R} \mid f_{n}(x)>a\right\} <+\infty$ .

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Problem IX.1.9

(a)(b)

Let $\quad \left\{ f_{n}\right\} \subset \mathscr {L}(\mathbb {R}) \quad$ and $\quad \varepsilon_{n} \rightarrow 0$ . Prove that if $\sum_{n \geqslant 1} \lambda \left\{ x \in \mathbb {R} \mid \left|f_{n}(x)\right| \geqslant \varepsilon_{n}\right\} <+\infty$ , then

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(a)

$f_{n}(x) \rightarrow 0$ almost everywhere on $\mathbb {R}$ ;

(b)

for any number $\varepsilon >0$ there is a set $e \subset \mathbb {R}$ such that $\lambda (e)<\varepsilon$ and $f_{n} \rightrightarrows 0$ on $\mathbb {R} \backslash e$ .

Problem IX.1.10

Prove that almost-everywhere convergence is "stable" in the following sense: if $\left\{ f_{n}\right\} \subset \mathscr {L}^{0}(0,1)$ and $f_{n}(x) \rightarrow 0$ almost everywhere on $(0,1)$ , then there exists a numerical sequence $\left\{ A_{n}\right\}$ such that $A_{n} \rightarrow \infty$ and $A_{n} f_{n}(x) \rightarrow 0$ almost everywhere on $(0,1)$ . Does this statement remain true if the interval $(0,1)$ is replaced by $\mathbb {R}$ ?

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Problem IX.1.11

Let $f, f_{n} \in \mathscr {L}^{0}(0,1), f_{n}(x) \rightarrow f(x)$ almost everywhere on (0,1). Prove that there exist a function $g \in \mathscr {L}^{0}(0,1)$ (a "regulator of convergence") and a numerical sequence $\left\{ \varepsilon_{n}\right\}$ , such that $\varepsilon_{n} \rightarrow 0$ and $\left|f_{n}(x)-f(x)\right| \leqslant \varepsilon_{n} g(x)$ almost everywhere on $(0,1)$ for any $n \in \mathbb {N}$ . Does this statement remain true if the interval $(0,1)$ is replaced by $\mathbb {R}$ ?

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Problem IX.1.12

Relying on the result of the previous problem, prove Egorov's theorem: if $f, f_{n} \in \mathscr {L}^{0}(0,1)$ and $f_{n}(x) \rightarrow f(x)$ almost everywhere on $(0,1)$ , then for any number $\varepsilon >0$ there is a set $e \subset (0,1)$ such that $\lambda (e)<\varepsilon$ and $f_{n} \rightrightarrows f$ on $(0,1) \backslash e$ . Does this statement remain true if the interval $(0,1)$ is replaced by $\mathbb {R}$ ?

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Problem IX.1.13

Let $\left\{ f_{n}\right\} \subset \mathscr {L}^{0}(0,1)$ . Prove that $f_{n} \rightarrow 0$ in measure if and only if for any subsequence $\left\{ f_{n_{k}}\right\}$ there is, in turn, a subsequence $\left\{ f_{n_{k_{j}}}\right\}$ , converging to zero almost everywhere on (0,1).

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Problem IX.1.14

Let $\left\{ f_{n}\right\} \subset \mathscr {L}^{0}(\mathbb {R}), g \in \mathscr {L}^{0}(\mathbb {R}),\left|f_{n}(x)\right| \leqslant g(x)$ almost everywhere on $\mathbb {R}$ for any $n \in \mathbb {N}$ . Prove that if the function $g$ satisfies the condition

\lambda \left\{ x \in \mathbb {R} \mid g(x)>\varepsilon \right\} <+\infty \quad \text{ for any } \varepsilon >0, \tag {*}

then convergence of the sequence $\left\{ f_{n}\right\}$ to zero almost everywhere on $\mathbb {R}$ implies that $f_{n} \rightarrow 0$ in measure. Is this true if condition (*) is violated?

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Problem IX.1.15

(a)(b)

Let $\left\{ f_{n, k}\right\}_{n, k} \geqslant 1 \subset \mathscr {L}^{0}(0,1),\left\{ f_{n}\right\} \subset \mathscr {L}^{0}(0,1), f_{0} \in \mathscr {L}^{0}(0,1)$ , and let $f_{n, k} \rightarrow f_{n}$ in measure, $f_{n} \rightarrow f_{0}$ in measure. Prove that there exists a strictly increasing sequence of indices $\left\{ k_{n}\right\}$ such that $f_{n, k_{n}} \rightarrow f_{0}$ in measure (the diagonal sequence theorem). Does an analogous statement hold if convergence in measure is replaced by

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(a)

convergence almost everywhere on (0,1);

(b)

pointwise convergence on $(0,1)$ ?

Problem IX.1.16

(a)(b)

Let $a_{n} \geqslant 0, \sum a_{n}<\infty , \Delta =(0,1)$ . Prove that

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(a)

if $\sum a_{n} \ln n<\infty$ , then the series

\sum \frac{a_{n}}{\left|x-x_{n}\right|}

converges almost everywhere on $\Delta$ (with respect to Lebesgue measure) for any sequence $\left\{ x_{n}\right\} \subset \mathbb {R}$ ;

(b)

if $\sum a_{n} \ln n=\infty$ and $X$ is a countable set everywhere dense in $\Delta$ , then, depending on the way of numbering the elements $\left\{ x_{n}\right\}$ of the set $X$ , the series (1) may either converge almost everywhere on $\Delta$ or diverge almost everywhere (or even everywhere) on $\Delta$ .

Section IX.2Convergence in the Mean. The Law of Large Numbers[20/90%]

Problem IX.2.1

(a)(b)

Let $f \in \mathscr {L}^{r}(E)$ . Prove that

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(a)

if $0<s<r$ , then $f \in \mathscr {L}^{s}(E)$ ;

(b)

if $\lambda_{m}(E)=1$ , then the function $s \mapsto \left\| f\right\|_{s}$ increases on ( $0, r$ ].

Problem IX.2.2

(a)(b)

Let $\quad \left\{ f_{n}\right\} , \quad \left\{ g_{n}\right\} \subset \mathscr {L}^{r}(E), \quad f_{0}, \quad g_{0} \in \mathscr {L}^{r}(E), \quad$ with $\left\| f_{n}-f_{0}\right\|_{r} \rightarrow 0$ and $\left\| g_{n}-g_{0}\right\|_{r} \rightarrow 0$ . Prove that

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(a)

$f_{n} \rightarrow f_{0}$ in measure;

(b)

$\left\| f_{n} g_{n}-f_{0} g_{0}\right\|_{r / 2} \rightarrow 0$ .

Problem IX.2.3

Let $\left\{ f_{n}\right\} \subset \mathscr {L}^{1}(E)$ . Prove that if $f_{n} \rightarrow 0$ in measure and $\sup_{n}\left\| f_{n}\right\|_{1}<\infty$ , then $\int_{E} \sqrt{\left|f_{n}(x) g(x)\right|} d x \rightarrow 0$ for any function $g \in \mathscr {L}^{1}(E)$ .

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Problem IX.2.4

Let $\left\{ f_{n}\right\} \subset \mathscr {L}^{2}(E), f_{0} \in \mathscr {L}^{2}(E)$ . Prove that if $f_{n} \rightarrow f_{0}$ in measure and $\left\| f_{n}\right\|_{2} \rightarrow \left\| f_{0}\right\|_{2}$ , then $\left\| f_{n}-f_{0}\right\|_{2} \rightarrow 0$ .

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Problem IX.2.5

(a)(b)(c)(d)

Let $f, g \in \mathscr {L}^{0}(E)$ and $0<\alpha <\lambda_{m}(E)$ . Set

\mathscr {J}_{\alpha }(f)=\inf \left\{ t \mid \lambda _{m}\left\{ x \in E \mid \left|f(x)\right|>t\right\} <\alpha \right\} .

The quantity $\mathscr {J}_{\alpha }(f)$ is called the quantile (compare the definition of $\mathscr {J}_{\alpha }(f)$ with the definition of the nonincreasing rearrangement of the function $f$ before problem VIII.2.13; the quantile $\mathscr {J}_{1 / 2}(f)$ is the median of the function $\left|f\right|$ — see problem VIII.2.12). Prove that

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(a)

$\mathscr {J}_{\alpha }(f) \leqslant \mathscr {J}_{\beta }(f)$ for $0<\beta <\alpha$ ;

(b)

$\mathscr {J}_{\alpha }(a f)=\left|a\right| \mathscr {J}_{\alpha }(f)$ for any $a \in \mathbb {R}$ ;

(c)

$\mathscr {J}_{\alpha }(f+g) \leqslant \mathscr {J}_{\alpha / 2}(f)+\mathscr {J}_{\alpha / 2}(g)$ ;

(d)

$f_{n} \rightarrow f_{0}$ in measure ( $f_{n}, f_{0} \in \mathscr {L}^{0}(E)$ ) if and only if $\mathscr {F}_{\alpha }\left(f_{n}-f_{0}\right) \rightarrow 0$ for any $\alpha , 0<\alpha <\lambda_{m}(E)$ .

Problem IX.2.6

(a)(b)(c)

Let $M \subset \mathscr {L}^{2}(E)$ and let $\left\| f\right\|_{2} \leqslant C\left\| f\right\|_{1}$ for any function $f \in$ M. Prove that

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(a)

if $0<r<1$ , then there exists a number $\theta =\theta (r)$ such that $\left\| f\right\|_{1}<C^{\theta }\left\| f\right\|_{r}$ for any function $f \in M$ ;

(b)

there exists a number $\alpha =\alpha (C)>0$ such that $\left\| f\right\|_{1} \leqslant 2 \lambda_{m}(E) \mathscr {J}_{\alpha }(f)$ for any function $f \in M$ ;

(c)

if $\left\{ f_{n}\right\} \subset M$ , then $\left\| f_{n}\right\|_{2} \rightarrow 0$ if and only if $f_{n} \rightarrow 0$ in measure.

Problem IX.2.7

Prove that if the series ${}^{*)}$ $\sum \left|a_{n} \sin \left(n x+\varphi_{n}\right)\right|$ converges on a set of positive measure, then $\sum \left|a_{n}\right|<\infty$ .

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Remark.

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${}^{*)}$ Recall that $\sum c_{n}$ is notation for the series $\sum_{n \geqslant 1} c_{n}$ .

Problem IX.2.8

Let $\left\{ f_{n}\right\} \subset \mathscr {L}^{2}(E)$ and $\left\| f_{n}\right\|_{2} \leqslant C,\left\| f_{n}\right\|_{1} \geqslant \delta$ for some $C$ , $\delta >0$ and any $n \in \mathbb {N}$ . Prove that if the series $\sum a_{n} f_{n}(x)$ converges absolutely almost everywhere on $E$ , then $\sum \left|a_{n}\right|<\infty$ .

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Problem IX.2.9

Let $\left\{ f_{n}\right\}$ be an orthogonal system in $\mathscr {L}^{2}(E), \sigma_{n}=\frac{1}{n} \sum_{1 \leqslant k \leqslant n} f_{k}$ . Prove that if $\frac{1}{n^{2}} \sum_{1 \leqslant k \leqslant n}\left\| f_{k}\right\|_{2}^{2} \rightarrow 0$ , then $\sigma_{n} \rightarrow 0$ in measure. (Various versions of this statement are called the law of large numbers.)

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Problem IX.2.10

(a)(b)(c)(d)

Prove that the indicated sequences $\left\{ f_{n}\right\}$ converge in measure on the interval $(-\pi , \pi )$ , and find their limits:

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(a)

$f_{n}(x)=\frac{1}{n} \sum_{1 \leqslant k \leqslant n} \sin^{2} k x$ ;

(b)

$f_{n}(x)=\frac{1}{n} \sum_{1 \leqslant k \leqslant n} \sqrt[3]{k} \sin k x$ ;

(c)

$f_{n}(x)=\frac{1}{n} \sum_{1 \leqslant k \leqslant n}\left(1+\frac{1}{k}\right)^{k} \sin^{2} k x$ ;

(d)

$f_{n}(x)=\frac{1}{n} \sum_{1 \leqslant k \leqslant n}\left(1-\frac{k}{n}\right) \sin^{2} k x$ .

Problem IX.2.11

Find the limit

\lim _{n \rightarrow \infty } \int _{-\pi }^{\pi } \operatorname {arctg}\left(\frac{1}{n} \sum _{1 \leqslant k \leqslant n} k \sin ^{2} k x\right) d x

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Problem IX.2.12

(a)(b)

Let $\left\{ f_{n}\right\}$ be an orthogonal system in $\mathscr {L}^{2}(E), \sigma_{n}=\frac{f_{1}+\ldots +f_{n}}{n}$ . Prove that

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(a)

$\sigma_{k^{2}}(x) \rightarrow 0$ almost everywhere on $E$ , if the norms $\left\| f_{n}\right\|_{2}$ are bounded;

(b)

$\sigma_{n}(x) \rightarrow 0$ almost everywhere on $E$ , if the norms $\left\| f_{n}\right\|_{\infty }$ are bounded. The latter statement is often formulated by saying that the strong law of large numbers holds for the sequence $\left\{ f_{n}\right\}$ .

Problem IX.2.13

Let $\left\{ f_{n}\right\}$ be an orthogonal system in $\mathscr {L}^{2}(E), \sigma_{n}=\frac{f_{1}+\ldots +f_{n}}{n}$ . Prove that if $\sum \frac{1}{n^{3 / 2}}\left\| f_{n}\right\|_{2}^{2}<\infty$ , then $\sigma_{n}(x) \rightarrow 0$ almost everywhere on $E$ , i.e., the strong law of large numbers holds for the sequence $\left\{ f_{n}\right\}$ . Prove, moreover, that the function $h=\sup_{n}\left|\sigma_{n}\right|$ belongs to $\mathscr {L}^{2}(E)$ .

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Problem IX.2.14

Let $\varepsilon_{n}(x)$ be the $n$ -th binary digit of the expansion of the number $x \in (0,1]$ , $\sigma_{n}(x)=\frac{1}{n}\left(\varepsilon_{1}(x)+\ldots +\varepsilon_{n}(x)\right)$ . Prove that $\sigma_{n}(x) \rightarrow \frac{1}{2}$ almost everywhere on (0,1), i.e., for almost all numbers of the interval (0,1) zeros and ones occur equally often in their binary expansions.

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Problem IX.2.15

Let $m$ be one of the digits $0, \ldots , 9$ , and let $c_{m}(x)=1$ if the decimal expansion of the number $x-\left[x\right]$ has the form $0, m \ldots$ , and $c_{m}(x)=0$ otherwise. Set $\sigma_{n}(x)=\frac{1}{n} \sum_{0 \leqslant k<n} c_{m}\left(10^{k} x\right)$ . Prove that $\sigma_{n}(x) \rightarrow \frac{1}{10}$ almost everywhere on $(0,1)$ .

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Problem IX.2.16

A number $x \in (0,1)$ is called normal if for any $p \in \mathbb {N}$ its expansion as a $p$ -ary fraction contains all "digits" (the numbers $0,1, \ldots , p-1$ ) equally often. It is known, for example, that the number

x=0,12345678910111213141516171819202122 \ldots

(the decimal expansion of $x$ consists of all natural numbers written one after another in decimal notation) is normal [Kats].

Generalizing the results of problems 2.14 and 2.15, prove that almost all numbers of the interval $(0,1)$ are normal.

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Problem IX.2.17

Let $f \in \mathscr {L}^{2}(0,1), \int_{0}^{1} f(x) d x=0$ . For arbitrary $n \in \mathbb {N}$ and $\varepsilon >0$ set

E_{n}(\varepsilon )=\left\{ \left(x_{1}, \ldots , x_{n}\right) \in (0,1)^{n} \mid \left|\frac{1}{n} \sum _{1 \leqslant x \leqslant n} f\left(x_{k}\right)\right|>\varepsilon \right\} .

Prove that $\lambda_{n}\left(E_{n}(\varepsilon )\right) \rightarrow 0$ as $n \rightarrow \infty$ .

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Problem IX.2.18

Let $\left\{ f_{n}\right\} \subset \mathscr {L}^{2}(0,1), \int_{0}^{1} f_{n}(x) d x=0$ for every $n \in \mathbb {N}$ . For arbitrary $n \in \mathbb {N}$ and $\varepsilon >0$ set

E_{n}(\varepsilon )=\left\{ \left(x_{1}, \ldots , x_{n}\right) \in (0,1)^{n} \mid \left|\frac{1}{n} \sum _{1 \leqslant k \leqslant n} f_{k}\left(x_{k}\right)\right|>\varepsilon \right\} .

Prove that $\lambda_{n}\left(E_{n}(\varepsilon )\right) \underset {n \rightarrow \infty }{\longrightarrow } 0$ , if $\frac{1}{n^{2}} \sum_{1 \leqslant k \leqslant n}\left\| f_{k}\right\|_{2}^{2} \rightarrow 0$ .

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Problem IX.2.19

Let $\varphi \in C(\mathbb {R}), \sup_{\mathbb {R}}\left|\varphi \right|<\infty$ . Find the limit

\lim _{n \rightarrow \infty } \int _{\mathbb {R}^{n}} \varphi \left(\frac{\left\| x\right\| ^{2}}{n}\right) e^{-\pi \left\| x\right\| ^{2}} d x

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Problem IX.2.20

(a)(b)(c)

Let $\left\{ a_{n}\right\} \subset \mathbb {C},\left\{ f_{n}\right\}$ be an orthonormal sequence in $\mathscr {L}^{2}(E), S_{n}=\sum_{1 \leqslant k \leqslant n} a_{k} f_{k}$ . Prove that if $\sum \sqrt{n}\left|a_{n}\right|^{2}<\infty$ , then

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(a)

$\sum \left\| S-S_{k^{2}}\right\|_{2}^{2}<\infty$ , where $S=\sum a_{k} f_{k}$ ;

(b)

the series $\sum a_{n} f_{n}(x)$ converges almost everywhere on $E$ ;

(c)

$\sup_{n}\left|S_{n}\right| \in \mathscr {L}^{2}(E)$ .

Section IX.3Rademacher Functions. The Khinchin Inequality[31/90%]

Problem IX.3.1

(a)(b)(c)(d)

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(a)

Prove that for any $n \in \mathbb {N}$ and any collection $\left\{ \varepsilon_{k}\right\}_{1 \leqslant k \leqslant n}$ , where the numbers $\varepsilon_{k}$ take the values +1 or -1, there exists a point $x_{0} \in (0,1)$ such that $r_{k}\left(x_{0}\right)=\varepsilon_{k}$ for $k=1, \ldots , n$ . What is the measure of the set of such points?

(b)

Prove that $r_{1}(x+1)=r_{1}(x), r_{n}(x)=r_{1}\left(2^{n-1} x\right)$ .

(c)

Let $a_{n}(x)$ be the $n$ -th digit of the binary expansion of the number $x \in (0,1)$ . Prove that $a_{n}(x)=\frac{1}{2}\left(1-r_{n}(x)\right)$ almost everywhere on (0,1).

(d)

Find the sum of the series $\sum 2^{-n} r_{n}(x)(x \in (0,1))$ .

Problem IX.3.2

(a)(b)

Prove that

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(a)

$\int_{0}^{1} \varphi \left(r_{1}(x), \ldots , r_{n}(x)\right) d x=2^{-n} \sum_{\varepsilon_{1}, \ldots , \varepsilon_{n} \in \left\{ -1,1\right\} } \varphi \left(\varepsilon_{1}, \ldots , \varepsilon_{n}\right)$ ;

(b)

$\int_{0}^{1}\left(\prod_{1 \leqslant k \leqslant n} \varphi_{k}\left(r_{k}(x)\right)\right) d x=\prod_{1 \leqslant k \leqslant n}\left(\int_{0}^{1} \varphi_{k}\left(r_{1}(x)\right) d x\right)$ .

Problem IX.3.3

(a)(b)

Let $m_{1}, \ldots , m_{n}$ be pairwise distinct natural numbers. Prove that

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(a)

$\int_{0}^{1} \varphi \left(r_{m_{1}}(x), r_{m_{2}}(x), \ldots , r_{m_{n}}(x)\right) d x=$

=\int _{0}^{1} \varphi \left(r_{1}(x), r_{2}(x), \ldots , r_{n}(x)\right) d x

(b)

$\int_{0}^{1} r_{m_{1}}(x) r_{m_{2}}(x) \ldots r_{m_{n}}(x) d x=0$ .

In particular, $\left\{ r_{n}\right\}$ is an orthonormal system in $\mathscr {L}^{2}(0,1)$ .

Problem IX.3.4

(a)(b)(c)(d)

Evaluate the integrals

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(a)

$\int_{0}^{1} e^{i t \Sigma 2^{-k} r_{k}(x)} d x$ ;

(b)

$\int_{0}^{1} e^{i t \Sigma 3^{-k} r_{k}(x)} d x$ ;

(c)

$\int_{0}^{1} e^{z\left(c_{1} r_{1}(x)+\ldots +c_{n} r_{n}(x)\right)} d x\left(z, c_{k} \in \mathbb {C}\right)$ .

(d)

Prove that if $t, a_{n} \in \mathbb {R}, a_{n} \rightarrow 0, \sum a_{n}^{2}=\infty$ , then

\int _{0}^{1} e^{i t\left(a_{1} r_{1}(x)+\ldots +a_{n} r_{n}(x)\right)} d x \underset {n \rightarrow \infty }{\longrightarrow } 0 \text{ for } t \neq 0

Problem IX.3.5

(a)(b)

Prove that for any complex numbers $c_{1}, \ldots , c_{n}$ the following inequalities hold:

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(a)

$\int_{0}^{1}\left|\sum_{1 \leqslant k \leqslant n} c_{k} r_{k}(x)\right|^{4} d x \leqslant A\left(\sum_{1 \leqslant k \leqslant n}\left|c_{k}\right|^{2}\right)^{2}$ ;

(b)

$\left(\sum_{1 \leqslant k \leqslant n}\left|c_{k}\right|^{2}\right)^{1 / 2} \leqslant B \int_{0}^{1}\left|\sum_{1 \leqslant k \leqslant n} c_{k} r_{k}(x)\right| d x$ ,

where $A, B$ are absolute constants.

Problem IX.3.6

(a)(b)

Prove that for any complex numbers $c_{k j}(k, j \in \mathbb {N})$ the following inequalities hold:

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(a)

$\int_{0}^{1} \int_{0}^{1}\left|\sum_{1 \leqslant k, j \leqslant n} c_{k j} r_{k}(x) r_{j}(y)\right|^{4} d x d y \leqslant A\left(\sum_{1 \leqslant k, j \leqslant n}\left|c_{k j}\right|^{2}\right)^{2}$ ;

(b)

$\left(\sum_{1 \leqslant k, j \leqslant n}\left|c_{k j}\right|^{2}\right)^{\frac{1}{2}} \leqslant B \int_{0}^{1} \int_{0}^{1}\left|\sum_{1 \leqslant k, j \leqslant n} c_{k j} r_{k}(x) r_{j}(y)\right| d x d y$ , where $A, B$ are absolute constants.

Problem IX.3.7

(a)(b)

Prove that

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(a)

$\left\{ r_{j} r_{k}\right\}_{j<k}$ is an orthonormal system in $\mathscr {L}^{2}(0,1)$ ;

(b)

for any complex numbers $c_{j k}$ the following inequalities hold:

\begin{aligned} & \int _{0}^{1}\left|\sum _{1 \leqslant j<k \leqslant n} c_{j k} r_{j}(x) r_{k}(x)\right|^{4} d x \leqslant A\left(\sum _{1 \leqslant j<k \leqslant n}\left|c_{j k}\right|^{2}\right)^{2} \\ & \left(\sum _{1 \leqslant j<k \leqslant n}\left|c_{j k}\right|^{2}\right)^{1 / 2} \leqslant B \int _{0}^{1}\left|\sum _{1 \leqslant j<k \leqslant n} c_{j k} r_{j}(x) r_{k}(x)\right| d x \end{aligned}

where $A, B$ are absolute constants.

Problem IX.3.8

(a)(b)

Prove that for any complex numbers $c_{1}, \ldots , c_{n}$ the following inequalities hold:

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(a)

$\int_{0}^{2 \pi }\left|\sum_{1 \leqslant k \leqslant n} c_{k} \sin 2^{k} x\right|^{4} d x \leqslant A\left(\sum_{1 \leqslant k \leqslant n}\left|c_{k}\right|^{2}\right)^{2}$ ;

(b)

$\left(\sum_{1 \leqslant k \leqslant n}\left|c_{k}\right|^{2}\right)^{1 / 2} \leqslant B \int_{0}^{2 \pi }\left|\sum_{1 \leqslant k \leqslant n} c_{k} \sin 2^{k} x\right| d x$ ,

where $A, B$ are absolute constants.

Problem IX.3.9

(a)(b)(c)

Relying on the results of problems $3.5-3.8$ , prove that for any $t>0$ and any complex numbers $c_{k}, c_{j k}$ the following inequalities hold:

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(a)

$\lambda \left\{ x \in (0,1) \mid \left|\sum_{k=1}^{n} c_{k} r_{k}(x)\right|>t\right\} \leqslant \frac{A_{1}}{t^{4}}\left(\sum_{k=1}^{n}\left|c_{k}\right|^{2}\right)^{2}$ ;

(b)

$\lambda \left\{ x \in (0,1) \mid \left|\sum_{1 \leqslant j<k \leqslant n} c_{j k} r_{j}(x) r_{k}(x)\right|>t\right\} \leqslant$

\leqslant \frac{A_{2}}{t^{4}}\left(\sum _{1 \leqslant j<k \leqslant n}\left|c_{j k}\right|^{2}\right)^{2} ;

(c)

$\lambda \left\{ x \in (0,2 \pi ) \mid \left|\sum_{k=1}^{n} c_{k} \sin 2^{k} x\right|>t\right\} \leqslant \frac{A_{3}}{t^{4}}\left(\sum_{k=1}^{n}\left|c_{k}\right|^{2}\right)^{2}$ , where $A_{1}, A_{2}, A_{3}$ are absolute constants.

Problem IX.3.10

(a)(b)

Let $\sigma_{n}(x)=\frac{1}{n} \sum_{1 \leqslant k \leqslant n} r_{k}(x)(x \in (0,1))$ . Prove that

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(a)

$\sum \lambda \left\{ x \in (0,1) \mid \sigma_{n}(x) \geqslant n^{-1 / 5}\right\} <\infty$ ;

(b)

$\sum \left|\sigma_{n}(x)\right|^{4}<\infty$ almost everywhere on $(0,1)$ .

Each of the statements a), b) implies the equality $\lim_{n \rightarrow \infty } \sigma_{n}(x)=0$ almost everywhere on (0,1), which is known as the strong law of large numbers. Using the connection between Rademacher functions and the digits of the binary expansion (see problem 3.1 c)), deduce from this that the binary expansion of almost every number contains "equally many" zeros and ones (see problem 2.14).

Problem IX.3.11

(a)(b)

Let $C$ be an arbitrary square $n \times n$ matrix, $A$ the set of all vectors in $\mathbb {R}^{n}$ whose coordinates equal $\pm 1$ .

Prove that

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(a)

$\sum_{x, y \in A}(C x, y)^{4} \leqslant K_{1} \cdot 2^{-2 n}\left(\sum_{x, y \in A}(C x, y)^{2}\right)^{2}$ ;

(b)

$\left(\sum_{x, y \in A}(C x, y)^{2}\right)^{1 / 2} \leqslant K_{2} \cdot 2^{-n} \sum_{x, y \in A}\left|(C x, y)\right|$ ,

where $K_{1}, K_{2}$ are absolute constants.

Problem IX.3.12

(a)(b)

Using the uniqueness theorem for the Fourier transform of finite Borel measures (see § 4), find the increasing distribution functions of the following functions:

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(a)

$f(x)=\frac{1}{2}+\sum 3^{-k} r_{k}(x)$ ;

(b)

$f(x)=\sum 2^{-k} r_{m_{k}}(x)$ ,

where $x \in (0,1)$ , and $\left\{ m_{k}\right\}$ is an arbitrary (not necessarily increasing) sequence of pairwise distinct natural numbers.

Problem IX.3.13

(a)(b)(c)

Let $R_{n}=a_{1} r_{1}+\ldots +a_{n} r_{n}\left(a_{1}, \ldots , a_{n} \in \mathbb {R}\right), \Delta$ be one of the intervals $\Delta_{m, j}$ (see the definition of the Rademacher functions). Prove that for $m \leqslant n$ the following inequalities hold:

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(a)

$\int_{\Delta }\left|R_{m}(x)\right| d x \leqslant \int_{\Delta }\left|R_{n}(x)\right| d x ;$

(b)

$\int_{\Delta } e^{R_{m}(x)} d x \leqslant \int_{\Delta } e^{R_{n}(x)} d x$ .

(c)

Prove that inequalities a), b) remain valid if $\Delta$ is replaced by the set $E=\left\{ x \in (0,1) \mid \left|\sum_{1 \leqslant k \leqslant m} a_{k} r_{k}(x)\right|>t\right\}$ .

Problem IX.3.14

(a)(b)

Let $R_{n}=a_{1} r_{1}+\ldots +a_{n} r_{n}, A^{2}=\left|a_{1}\right|^{2}+\ldots +\left|a_{n}\right|^{2}\left(a_{1}, a_{2}, \ldots , a_{n} \in \mathbb {R}\right)$ . Using the inequality $\operatorname {ch} u \leqslant e^{u^{2} / 2}(u \in \mathbb {R})$ , prove that for $t>0$

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(a)

$\lambda \left\{ x \in (0,1) \mid \left|R_{n}(x)\right|>t\right\} \leqslant 2 e^{-t^{2} /\left(2 A^{2}\right)}$ ;

(b)

if $E=\left\{ x \in (0,1) \mid \max \left\{ \left|R_{1}(x)\right|, \ldots ,\left|R_{n}(x)\right|\right\} >t\right\}$ , then $\lambda (E) \leqslant \frac{A}{t}$ , $\lambda (E) \leqslant 2 e^{-t^{2} /\left(2 A^{2}\right)}$ .

Verify that for complex $a_{k}$ inequality a) is replaced by $\lambda \left\{ \ldots \right\} \leqslant 4 e^{-t^{2} /\left(4 A^{2}\right)}$ .

Problem IX.3.15

(a)(b)

Let $\sum \left|a_{k}\right|^{2}<\infty$ . Prove that

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(a)

the series $\sum a_{k} r_{k}(x)$ converges almost everywhere;

(b)

if $M(x)=\sup_{n \geqslant 1}\left|a_{1} r_{1}(x)+\ldots +a_{n} r_{n}(x)\right|$ , then $\int_{0}^{1} e^{s M^{2}(x)} d x<\infty$ for any number $s>0$ .

Problem IX.3.16

Prove that if the series $\sum a_{k} r_{k}$ converges in measure on some set $E$ of positive measure, then $\sum \left|a_{k}\right|^{2}<\infty$ .

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Problem IX.3.17

Prove that the following statements are equivalent:

$\sum \left|a_{k}\right|^{2}<\infty$ ;
the series $\sum a_{k} r_{k}$ converges in $\mathscr {L}^{p}(0,1)$ for some $p \in (0,2)$ ;
the series $\sum a_{k} r_{k}$ converges in measure on the interval $(0,1)$ ;
the series $\sum a_{k} r_{k}$ converges almost everywhere on $(0,1)$ .

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Problem IX.3.18

Prove that if the series $\sum a_{k} \sin 2^{k} x$ converges in measure on some set of positive measure, then $\sum \left|a_{k}\right|^{2}<\infty$ .

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Problem IX.3.19

Let $\sum \left|a_{k}\right|^{2}<\infty , \Delta$ be a nonempty interval. Prove that if vrai sup $\left|\sum a_{k} r_{k}(x)\right|<\infty$ , then $\sum \left|a_{k}\right|<\infty$ (cf. problem 4.15). $x \in \Delta$

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Problem IX.3.20

(a)(b)

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(a)

Let $\left\{ \varphi_{n}\right\}_{n \geqslant 1}$ be an orthonormal system in the space $\mathscr {L}^{2}(0,1)$ . Prove that

\sum _{n \geqslant 1}\left|\int _{e} \varphi _{n}(t) d t\right|^{2} \leqslant \lambda (e) \quad \text{ for } \quad e \subset (0,1) .

(b)

Sharpen this inequality for the Rademacher system.

Problem IX.3.21

(a)(b)(c)(d)(e)(f)

The Haar system of functions $\left\{ h_{m}\right\}_{m \geqslant 0}$ is defined as follows. Every natural number $m$ can be uniquely represented in the form $m=2^{n}+k$ , where $n, k$ are nonnegative integers, $0 \leqslant k<2^{n}$ . By definition we set $h_{0}(x)=1$ for any $x \in (0,1)$ , and for $m \in \mathbb {N}$

h_{m}(x)= \begin{cases} 2^{n / 2}, & \text{ for } 2^{-n} k<x<2^{-n}\left(k+\frac{1}{2}\right), \\ -2^{n / 2}, & \text{ for } 2^{-n}\left(k+\frac{1}{2}\right)<x<2^{-n}(k+1), \\ 0, & \text{ for all other values of } x \in (0,1) . \end{cases}

Prove that

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(a)

$\left\{ h_{m}\right\}_{m \geqslant 0}$ is an orthonormal system in $\mathscr {L}^{2}(0,1)$ ;

(b)

if $E=\left\{ x \in (0,1) \mid \left|\sum_{0 \leqslant k \leqslant m} a_{k} h_{k}(x)\right|>t\right\}$ , then for $n \geqslant m$

\int _{E}\left|\sum _{0 \leqslant k \leqslant m} a_{k} h_{k}(x)\right| d x \leqslant \int _{E}\left|\sum _{0 \leqslant k \leqslant n} a_{k} h_{k}(x)\right| d x ;

(c)

if $E=\left\{ x \in (0,1) \mid \max_{0 \leqslant m \leqslant n}\left|\sum_{0 \leqslant k \leqslant m} a_{k} h_{k}(x)\right|>t\right\}$ , then

\lambda (E) \leqslant A t^{-1}, \quad \text{ where } \quad A^{2}=\left|a_{1}\right|^{2}+\ldots +\left|a_{n}\right|^{2} ;

(d)

if $\sum \left|a_{k}\right|^{2}<\infty$ , then the series $\sum a_{k} h_{k}(x)$ converges almost everywhere on (0,1);

(e)

the partial sum $S_{2^{n}}$ of the Fourier series of a function $f$ from $\mathscr {L}^{1}(0,1)$ with respect to the Haar system is the result of averaging $f$ over the intervals $\Delta_{n, k}=\left(\frac{k}{2^{n}}, \frac{k+1}{2^{n}}\right), k=0,1, \ldots , 2^{n}-1$ ;

(f)

deduce from e) that $\left\| S_{2^{n}}\right\|_{1} \leqslant \left\| f\right\|_{1}$ .

Problem IX.3.22

(a)(b)

h_{m}(x)= \begin{cases} 2^{n / 2}, & \text{ for } 2^{-n} k<x<2^{-n}\left(k+\frac{1}{2}\right), \\ -2^{n / 2}, & \text{ for } 2^{-n}\left(k+\frac{1}{2}\right)<x<2^{-n}(k+1), \\ 0, & \text{ for all other values of } x \in (0,1) . \end{cases}

It is known that the Fourier series with respect to the Haar system of any summable function converges to it ${ }^{*)}$ in $\mathscr {L}^{1}(0,1)$ . Relying on this, prove that

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(a)

this series converges almost everywhere on $(0,1)$ ;

(b)

if $E=\left\{ x \in (0,1) \mid \sup_{n}\left|S_{n}(x)\right|>t\right\}$ , where $S_{n}$ is the $n$ -th partial sum of the Fourier series of the function $f$ with respect to the Haar system, then $\lambda (E) \leqslant \frac{\left\| f\right\|_{1}}{t}$ .

Remark.

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${ }^{*)}$ A reader familiar with the elements of functional analysis will notice that this is easily derived from the Banach-Steinhaus theorem.

Problem IX.3.23

(a)(b)

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(a)

Using the result of Problem 3.14, prove that for any $p>0$ and any $a_{1}, \ldots , a_{n} \in \mathbb {R}$ the Khintchine inequality holds:

A_{p}\left(\sum _{1 \leqslant k \leqslant n} a_{k}^{2}\right)^{1 / 2} \leqslant \left\| \sum _{1 \leqslant k \leqslant n} a_{k} r_{k}\right\| _{p} \leqslant B_{p}\left(\sum _{1 \leqslant k \leqslant n} a_{k}^{2}\right)^{1 / 2},

where $A_{p}, B_{p}$ are positive constants depending only on $p$ , and $\varlimsup_{p \rightarrow +\infty } B_{p} / \sqrt{p} \leqslant 1 / \sqrt{e}$ .

Note that the lower bound for $p \geqslant 2$ is obvious (with coefficient $A_{p}=1$ , which cannot be increased), while the upper bound is obvious for $p \leqslant 2$ (with coefficient $B_{p}=1$ , which cannot be decreased).

(b)

Prove that $\quad A_{p} \leqslant \alpha_{p}=2^{\frac{1}{2}-\frac{1}{p}} \min \left\{ 1, \gamma_{p}\right\} \quad$ for $\quad p \in (0,2]$ , $B_{p} \geqslant \beta_{p}=2^{\frac{1}{2}-\frac{1}{p}} \gamma_{p}$ for $p \geqslant 2$ , where $\gamma_{p}=\left(\frac{\Gamma \left(\frac{1+p}{2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right)^{\frac{1}{p}}(p>0)$ (it is proved in [Haag] that $\alpha_{p}, \beta_{p}$ are the best values of the coefficients $A_{p}$ and $B_{p}$ ). In particular, $B_{2 m} \geqslant ((2 m-1)!!)^{1 /(2 m)}$ for all $m \in \mathbb {N}$ .

Problem IX.3.24

Prove that if for $p>2$ and some $B_{p}$ the inequality

\left\| \sum _{1 \leqslant k \leqslant n} a_{k} r_{k}\right\| _{p} \leqslant B_{p}\left(\sum _{1 \leqslant k \leqslant n}\left|a_{k}\right|^{2}\right)^{1 / 2}

holds for any real numbers $a_{1}, \ldots , a_{n}$ , then it also holds for any complex $a_{1}, a_{2}, \ldots , a_{n}$ .

For $0<p<2$ an analogous statement holds for the inequality

\left\| \sum _{1 \leqslant k \leqslant n} a_{k} r_{k}\right\| _{p} \geqslant A_{p}\left(\sum _{1 \leqslant k \leqslant n}\left|a_{k}\right|^{2}\right)^{1 / 2} .

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Problem IX.3.25

(a)(b)

Let $p>0, n \in \mathbb {N}$ and $a_{1}, \ldots , a_{n} \in \mathbb {R}$ . Prove that

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(a)

$\left\| \sum_{k=1}^{n} a_{k} r_{k}\right\|_{p+2}^{p+2}=(p+1) \sum_{j=1}^{n} a_{j}^{2} \int_{0}^{1}\left\| t a_{j} r_{j}+\sum_{\substack {k=1 \\ k \neq j}}^{n} a_{k} r_{k}\right\|_{p}^{p} d t$ ;

(b)

the best constants $B_{p}$ in the Khintchine inequality satisfy the inequality $B_{p+2}^{p+2} \leqslant (p+1) B_{p}^{p}$ ; moreover, $B_{2 m}^{2 m}=(2 m-1)!!$ .

Problem IX.3.26

(a)(b)(c)

Let $n \in \mathbb {N}$ and $a_{1}, \ldots , a_{n} \in \mathbb {R}$ . Prove that

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(a)

$\int_{0}^{1}\left|a_{1} r_{1}(x)+\ldots +a_{n} r_{n}(x)\right| d x=\frac{2}{\pi } \int_{0}^{\infty }\left(1-\prod_{k=1}^{n} \cos a_{k} t\right) \frac{d t}{t^{2}}$ ;

(b)

$\int_{0}^{1}\left|a_{1} r_{1}(x)+\ldots +a_{n} r_{n}(x)\right| d x \geqslant \frac{1}{\sqrt{2}}\left(a_{1}^{2}+\ldots +a_{n}^{2}\right)^{1 / 2}$ ;

(c)

for $p \in (0,1]$ the best constant $A_{p}$ in the Khintchine inequality equals $2^{\frac{1}{2}-\frac{1}{p}}$ .

Problem IX.3.27

Prove that statements a)--d) of Problem 3.17 are equivalent to the statement: e) the series $\sum a_{k} r_{k}$ converges in $\mathscr {L}^{p}(0,1)$ for every $p$ in $(0, \infty )$ .

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Problem IX.3.28

(a)(b)

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(a)

Prove that for any trigonometric polynomial $\sum_{\left|k\right| \leqslant n} c_{k} e^{i k x}$ one can choose numbers $\varepsilon_{k}= \pm 1 \quad (\left|k\right| \leqslant n)$ such that

\int _{-\pi }^{\pi }\left|\sum _{\left|k\right| \leqslant n} \varepsilon _{k} c_{k} e^{i k x}\right| d x \geqslant c\left(\sum _{\left|k\right| \leqslant n}\left|c_{k}\right|^{2}\right)^{1 / 2},

where $c>0$ is an absolute constant.

(b)

Consider the trigonometric polynomial $T_{n}(\varphi )=\sum_{k=1}^{n} a_{k} e^{i k \varphi }$ , whose coefficients $a_{1}, \ldots , a_{n}$ form a permutation of the numbers $1,2, \ldots , n$ . Prove that the trivial estimate

\left\| T_{n}\right\| _{1}=\int _{-\pi }^{\pi }\left|T_{n}(\varphi )\right| d \varphi \leqslant \sqrt{2 \pi }\left\| T_{n}\right\| _{2}=2 \pi \sqrt{1^{2}+\ldots +n^{2}} \leqslant \operatorname {const} n^{3 / 2},

cannot, in general, be improved in order: for every $n \in \mathbb {N}$ there exists a permutation $a_{1}, \ldots , a_{n}$ of the numbers $1, \ldots , n$ such that $\left\| T_{n}\right\|_{1} \geqslant c n^{3 / 2}(c>0$ and independent of $n)$ .

Problem IX.3.29

Prove that for any trigonometric polynomial $\sum_{\left|k\right| \leqslant n} c_{k} e^{i k x}(n \geqslant 2)$ one can choose numbers $\varepsilon_{k}= \pm 1(\left|k\right| \leqslant n)$ such that

\sup _{x \in \mathbb {R}}\left|\sum _{\left|k\right| \leqslant n} \varepsilon _{k} c_{k} e^{i k x}\right| \leqslant 10 \sqrt{\ln n}\left(\sum _{\left|k\right| \leqslant n}\left|c_{k}\right|^{2}\right)^{1 / 2} .

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Problem IX.3.30

Prove that for any $a>2$ there exist numbers $q_{n}>0$ such that the series

\sum _{n \geqslant 2}\left(\frac{\left|r_{1}(x)+\ldots +r_{n}(x)\right|}{\sqrt{a n \ln n}}\right)^{q_{n}}

converges almost everywhere. Deduce from this that almost everywhere

\varlimsup _{n \rightarrow \infty } \frac{\left|r_{1}(x)+\ldots +r_{n}(x)\right|}{\sqrt{2 n \ln n}} \leqslant 1 .

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Problem IX.3.31

Let $\left\{ n_{j}\right\}$ be an arbitrary increasing sequence of natural numbers. Prove that

\varlimsup _{j \rightarrow +\infty } \frac{\left|r_{1}(x)+r_{2}(x)+\ldots +r_{n_{j}}(x)\right|}{\sqrt{2 n_{j} \ln j}} \leqslant 1

almost everywhere on $(0,1)$ . If $j=O\left(\ln n_{j}\right)$ , then we obtain from this a rather weak version of the law of the iterated logarithm:

\varlimsup _{j \rightarrow +\infty } \frac{\left|r_{1}(x)+r_{2}(x)+\ldots +r_{n_{j}}(x)\right|}{\sqrt{2 n_{j} \ln \ln n_{j}}} \leqslant 1 \text{ almost everywhere on }(0,1) \text{. }

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Section IX.4Fourier Series and Transform[57/96%]

Problem IX.4.1

Let $\left|\alpha \right|<1$ and $f_{\alpha }(x)=\left|x\right|^{\alpha }$ for $0<\left|x\right|<\pi$ . Find the asymptotics of the Fourier coefficients of the function $f_{\alpha }$ .

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Problem IX.4.2

(a)(b)

Let $f \in \mathscr {L}^{1}(0,2 \pi )$ . Prove that

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(a)

if $f$ is decreasing, then its sine Fourier coefficients are nonnegative;

(b)

if $f$ is convex, then its cosine Fourier coefficients are nonnegative.

Problem IX.4.3

Using the result of Problem IV.6.20, prove that the Fourier coefficients of a summable function can tend to zero arbitrarily slowly.

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Problem IX.4.4

Using the cosine Fourier coefficients of an even function, give an example of a nonzero series $\sum a_{n}$ such that $\sum_{j \geqslant 1} a_{k j}=0$ for any $k \in \mathbb {N}$ (cf. Problem IV.3.8).

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Problem IX.4.5

(a)(b)

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(a)

Let $0<\alpha <1, \quad f \in C(\left[-\pi , \pi \right])$ . Prove that if $\widehat{f}(n)=O\left(\left|n\right|^{-(1+\alpha )}\right)$ , then $f \in \operatorname {Lip}_{\alpha }$ . Is this true for $\alpha =1$ ?

(b)

Prove that if $0<\alpha \leqslant 1$ and $f$ is a $2 \pi$ -periodic function belonging to the class $\operatorname {Lip}_{\alpha }$ , then $\widehat{f}(n)=O\left(\left|n\right|^{-\alpha }\right)$ . Verify that this estimate is sharp.

Problem IX.4.6

Let $p \in (0,1)$ ,

f_{p}(x)=\sum _{n \geqslant 1} \frac{\cos n x}{n^{p}}, \quad g_{p}(x)=\sum _{n \geqslant 1} \frac{\sin n x}{n^{p}}

and let $h_{p}$ be the $2 \pi$ -periodic function defined on the interval $(0,2 \pi )$ by the equality

h_{p}(x)=\lim _{N \rightarrow \infty }\left(\left(\frac{x}{2 \pi }\right)^{p-1}+\left(1+\frac{x}{2 \pi }\right)^{p-1}+\ldots +\left(N+\frac{x}{2 \pi }\right)^{p-1}-\frac{N^{p}}{p}\right) .

Verify that $h_{p}$ is a linear combination of the functions $f_{p}$ and $g_{p}$ . Deduce from this that on the interval $(0, \pi )$ the functions $f_{p}, g_{p}$ are decreasing, convex, and

f_{p}(x)=\frac{A_{p}}{x^{1-p}}+O(1), \quad g_{p}(x)=\frac{B_{p}}{x^{1-p}}+O(1),

where $A_{p}, B_{p}$ are certain positive coefficients.

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Problem IX.4.7

(a)(b)

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(a)

Prove that if $f$ is a function of bounded variation on the interval $\left[-\pi , \pi \right]$ , then $\widehat{f}(n)=O\left(n^{-1}\right)$ .

(b)

Let $f(x)=x \cos \frac{1}{x}$ for $0<\left|x\right| \leqslant \pi , f(0)=0$ . Prove that $\widehat{f}(n)=O\left(n^{-1}\right)$ , although the variation of the function $f$ on $\left[-\pi , \pi \right]$ is infinite.

Problem IX.4.8

Let $f \in \mathscr {L}^{\infty }(-\pi , \pi )$ . Prove that the numbers $\widehat{f}(n) / \sqrt{\left|n\right|}(n \in \mathbb {Z}$ , $n \neq 0$ ) are the Fourier coefficients of a function belonging to the class $\operatorname {Lip}_{1 / 2}$ (use the construction of Problem III.3.23).

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Problem IX.4.9

Using Parseval's equality, prove Wirtinger's inequality ${ }^{*}$ ): if $f \in C^{1}(\left[0,2 \pi \right])$ , where $\widehat{f}(0)=0$ and $f(0)=f(2 \pi )$ , then

\int _{0}^{2 \pi }\left|f(x)\right|^{2} d x \leqslant \int _{0}^{2 \pi }\left|f^{\prime }(x)\right|^{2} d x

For which functions does this inequality become an equality?

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Problem IX.4.10

Let $f \in \mathscr {L}^{1}(-\pi , \pi )$ . Prove that $\sum_{n \in \mathbb {Z}}\left|\widehat{f}(n)\right|<\infty$ if and only if $f$ can be represented as the convolution of two $2 \pi$ -periodic functions from $\mathscr {L}^{2}(-\pi , \pi )$ .

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Problem IX.4.11

(a)(b)

For which $p>0$ do the series

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(a)

$\sum \frac{\sqrt[3]{\cos (2 \pi n \sqrt{2})}}{n^{p}}$ ;

(b)

$\sum (-1)^{n} \frac{\left|\sin (2 \pi n \sqrt{2})\right|}{n^{P}}$

converge?

*) For a proof not using Fourier series, see the hint to problem I.2.40. Various generalizations of the Wirtinger inequality can be found in [KhLP] on pp. 221, 384.

Problem IX.4.12

(a)(b)

Prove the convergence of the series

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(a)

$\sum \frac{\sqrt[3]{\cos \left(\pi \sqrt{2} n^{2}\right)}}{n}$ ;

(b)

$\sum (-1)^{n} \frac{\left|\sin \left(\pi \sqrt{2} n^{3}\right)\right|}{n}$ .

Problem IX.4.13

Let $S_{n}(x)=\sin x+\frac{1}{2} \sin 2 x+\ldots +\frac{1}{n} \sin n x-n$ -th partial sum of the Fourier series of the function $f(x)=\frac{1}{2}(\pi -x) \quad (x \in (0,2 \pi ))$ , $G_{n}=\max_{x} S_{n}(x)$ . Prove that the sums $S_{n}(x)$ are positive on $(0, \pi )$ and that

G_{n}=S_{n}\left(\frac{\pi }{n+1}\right) \uparrow G=\int _{0}^{\pi } \frac{\sin u}{u} d u>\frac{\pi }{2}=f(+0)

(Gibbs phenomenon).

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Problem IX.4.14

(a)(b)

Let $f(x)=\sum \frac{1}{n^{n}} \sin \left(n^{n^{n}}\left|x\right|\right)(\left|x\right| \leqslant \pi )$ . Prove that

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(a)

the Fourier series of the function $f$ diverges at the points $x=0, x= \pm \pi$ and converges at the remaining points of the interval $\left[-\pi , \pi \right]$ ;

(b)

if the continuous function $g$ coincides with $f$ on the interval $\left[a, \pi \right]$ , and is linear on the interval $\left[-\pi , a\right]$ , then its Fourier series diverges at the point $a$ for almost all $a \in (-\pi , \pi )$ .

Problem IX.4.15

Prove that the following three properties of the trigonometric series $\sum \left(a_{n} \cos 2^{n} x+b_{n} \sin 2^{n} x\right)$ are equivalent:

$\sum \left(\left|a_{n}\right|+\left|b_{n}\right|\right)<+\infty$ ;
the partial sums $S_{N}(x)=\sum_{n=0}^{N}\left(a_{n} \cos 2^{n} x+b_{n} \sin 2^{n} x\right)$ are uniformly bounded on some interval $\left[a, b\right], a<b$ ;
the sums $S_{N}(x)$ are uniformly bounded on $\mathbb {R}$ .

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Problem IX.4.16

(a)(b)

We will call a function $f$ of the form

f(x)=\sum _{k=1}^{n} a_{k} \cos \left(2 \pi m_{k} x+\varphi _{k}\right) \quad (x \in \mathbb {R}),

where $m_{k}, \varphi_{k}, a_{k} \in \mathbb {R}, m_{k} \geqslant 0$ , a real quasi-trigonometric polynomial. We will call the set $\left\{ m_{1}, \ldots , m_{n}\right\}$ the spectrum of the quasi-trigonometric polynomial $f$ , and the numbers $m_{k}$ its frequencies. Products of the form $\prod_{k=1}^{n}\left(1+\cos \left(2 \pi m_{k} x+\varphi_{k}\right)\right)$ ( $n \in \mathbb {N}$ ) are called Riesz products. A sequence $\left\{ m_{k}\right\} \subset \mathbb {R}_{+}$ will be called lacunary if it satisfies the condition

m_{k+1} / m_{k} \geqslant Q>1 \quad \text{ for all } k \in \mathbb {N} . \tag {*}

Let $\left\{ m_{k}\right\}$ be a sequence of positive numbers satisfying condition (*) for some $Q>1$ . Prove that

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(a)

if $Q>3$ , then for any natural number $n$ the Riesz product $R_{n}(x)=\prod_{k=1}^{n}\left(1+\cos \left(2 \pi m_{k} x+\varphi_{k}\right)\right)$ is a quasi-trigonometric polynomial of the form

R_{n}(x)=1+\sum _{k=1}^{n} \cos \left(2 \pi m_{k} x+\varphi _{k}\right)+\sum _{j=1}^{s} c_{j} \cos \left(2 \pi n_{j} x+\psi _{j}\right)

where $n_{j} \neq m_{k}$ for all $j=1, \ldots , s, k \in \mathbb {N}$ ;

(b)

there exists a natural number $r=r(Q)$ such that for any integer $p, 0 \leqslant p<r$ , and any $n, n \in \mathbb {N}$ , the Riesz product $\widetilde{R}_{n}(x)=\prod_{k=1}^{n}\left(1+\cos \left(2 \pi m_{k r+p} x+\widetilde{\varphi }_{k}\right)\right)$ contains no terms with frequencies $m_{j}$ different from $m_{r+p}, m_{2 r+p}, \ldots , m_{n r+p}$ , i.e. has the form

\widetilde{R}_{n}(x)=1+\sum _{k=1}^{n} \cos \left(2 \pi m_{k r+p} x+\widetilde{\varphi }_{k}\right)+\sum _{j=1}^{\widetilde{s}} \widetilde{c}_{j} \cos \left(2 \pi \widetilde{n}_{j} x+\widetilde{\psi }_{j}\right)

where $\widetilde{n}_{j} \neq m_{k}$ for all $j=1, \ldots , \widetilde{s}, k \in \mathbb {N}$ .

Problem IX.4.17

We call a function $f$ of the form $f(x)=\sum_{k=1}^{n} a_{k} \cos \left(2 \pi m_{k} x+\varphi_{k}\right)$ ( $x \in \mathbb {R}$ ), where $m_{k}, \varphi_{k}, a_{k} \in \mathbb {R}, m_{k} \geqslant 0$ , a real quasi-trigonometric polynomial; the numbers $m_{k}$ are called its frequencies. A sequence $\left\{ m_{k}\right\} \subset \mathbb {R}_{+}$ will be called lacunary if it satisfies the condition

m_{k+1} / m_{k} \geqslant Q>1 \quad \text{ for all } k \in \mathbb {N} . \tag {*}

Let $F(x)=\sum_{k=1}^{N} a_{k} \cos \left(2 \pi m_{k} x+\varphi_{k}\right)$ be a real quasi-trigonometric polynomial whose frequencies form a lacunary sequence satisfying condition (*), $M=\sup_{x \in \mathbb {R}}\left|F(x)\right|$ . Prove that there exists a number $C=C(Q)$ such that $\sum_{k=1}^{N}\left|a_{k}\right| \leqslant C M$ , whatever the coefficients $a_{k}$ may be.

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Problem IX.4.18

The Cesàro-Fejér sum of a function $f \in \mathscr {L}^{1}(-\pi , \pi )$ is the trigonometric polynomial

\sigma _{n}(f, x)=\sigma _{n}(x)=\frac{S_{0}(x)+S_{1}(x)+\ldots +S_{n-1}(x)}{n} \quad (x \in \mathbb {R}),

where $S_{0}, S_{1}, \ldots , S_{n}$ are the partial sums of the Fourier series of $f$ . Cesàro-Fejér sums can be represented as the convolution of the function $f$ and the Fejér kernel $\Phi_{n}$ :

\sigma _{n}(f, x)=\int _{-\pi }^{\pi } \Phi _{n}(x-y) f(y) d y, \quad \text{ where } \quad \Phi _{n}(x)=\frac{1}{2 \pi n} \frac{\sin ^{2}(n x / 2)}{\sin ^{2}(x / 2)};

moreover, $\sigma_{n}(f, x)=\sum_{\left|k\right|<n}\left(1-\frac{\left|k\right|}{n}\right) \widehat{f}(k) e^{i k x}$ .

Let $T$ be a trigonometric polynomial of order at most $n$ . Prove that

T^{\prime }=-2 n T * \Psi _{n}, \quad \text{ where } \Psi _{n}(x)=\Phi _{n}(x) \sin n x .

Deduce from this the weakened Bernstein inequality:

\max _{x}\left|T^{\prime }(x)\right| \leqslant 2 n \max _{x}\left|T(x)\right|

and analogous estimates for $L^{p}$ -norms (regarding the Bernstein inequality see problem I.2.22).

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Problem IX.4.19

(a)(b)(c)(d)

The Cesàro-Fejér sum of a function $f \in \mathscr {L}^{1}(-\pi , \pi )$ is the trigonometric polynomial

\sigma _{n}(f, x)=\sigma _{n}(x)=\frac{S_{0}(x)+S_{1}(x)+\ldots +S_{n-1}(x)}{n} \quad (x \in \mathbb {R}),

\sigma _{n}(f, x)=\int _{-\pi }^{\pi } \Phi _{n}(x-y) f(y) d y, \quad \text{ where } \quad \Phi _{n}(x)=\frac{1}{2 \pi n} \frac{\sin ^{2}(n x / 2)}{\sin ^{2}(x / 2)};

moreover, $\sigma_{n}(f, x)=\sum_{\left|k\right|<n}\left(1-\frac{\left|k\right|}{n}\right) \widehat{f}(k) e^{i k x}$ .

Let $f$ be a $2 \pi$ -periodic function, summable on $(-\pi , \pi )$ . Prove that

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(a)

if $f$ is continuous at the point $x_{0}$ , then $\sigma_{n}\left(x_{0}\right) \rightarrow f\left(x_{0}\right)$ , and if $f$ is continuous everywhere, then $\sigma_{n} \rightrightarrows f$ on $\mathbb {R}$ ;

(b)

if $f$ is continuous at all points of a compact set $\Delta$ , then $\max_{x \in \Delta }\left|\sigma_{n}(x)-f(x)\right| \rightarrow 0$ ;

(c)

if at the point $x$ finite one-sided derivatives $f_{+}^{\prime }(x)$ and $f_{-}^{\prime }(x)$ exist, then

\frac{n}{\ln n}\left(\sigma _{n}(x)-f(x)\right) \rightarrow \frac{1}{\pi }\left(f_{+}^{\prime }(x)-f_{-}^{\prime }(x)\right) ;

(d)

the convergence of $\left\{ \sigma_{n}\right\}$ to $f$ cannot be too fast: if $\lim n\left\| f-\sigma_{n}\right\|_{1}=0$ , then $f \equiv$ const almost everywhere.

Problem IX.4.20

(a)(b)(c)

The Cesàro-Fejér sum of a function $f \in \mathscr {L}^{1}(-\pi , \pi )$ is the trigonometric polynomial

\sigma _{n}(f, x)=\sigma _{n}(x)=\frac{S_{0}(x)+S_{1}(x)+\ldots +S_{n-1}(x)}{n} \quad (x \in \mathbb {R}),

\sigma _{n}(f, x)=\int _{-\pi }^{\pi } \Phi _{n}(x-y) f(y) d y, \quad \text{ where } \quad \Phi _{n}(x)=\frac{1}{2 \pi n} \frac{\sin ^{2}(n x / 2)}{\sin ^{2}(x / 2)};

moreover, $\sigma_{n}(f, x)=\sum_{\left|k\right|<n}\left(1-\frac{\left|k\right|}{n}\right) \widehat{f}(k) e^{i k x}$ .

Let $f$ be a $2 \pi$ -periodic function, $f \in \operatorname {Lip}_{\alpha }, 0<\alpha <1$ . Prove that as $n \rightarrow \infty$

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(a)

$\left\| \sigma_{n}-f\right\|_{\infty }=O\left(n^{-\alpha }\right)$ ;

(b)

$\left\| S_{n}-f\right\|_{2}=O\left(n^{-\alpha }\right)$ ;

(c)

$\left\| S_{n}-f\right\|_{\infty }=O\left(n^{-\alpha } \ln n\right)$ .

Problem IX.4.21

(a)(b)(c)

The Cesàro-Fejér sum of a function $f \in \mathscr {L}^{1}(-\pi , \pi )$ is the trigonometric polynomial

\sigma _{n}(f, x)=\sigma _{n}(x)=\frac{S_{0}(x)+S_{1}(x)+\ldots +S_{n-1}(x)}{n} \quad (x \in \mathbb {R}),

\sigma _{n}(f, x)=\int _{-\pi }^{\pi } \Phi _{n}(x-y) f(y) d y, \quad \text{ where } \quad \Phi _{n}(x)=\frac{1}{2 \pi n} \frac{\sin ^{2}(n x / 2)}{\sin ^{2}(x / 2)};

moreover, $\sigma_{n}(f, x)=\sum_{\left|k\right|<n}\left(1-\frac{\left|k\right|}{n}\right) \widehat{f}(k) e^{i k x}$ .

Let $f$ be a $2 \pi$ -periodic function, $f \in \operatorname {Lip}_{1}$ . Prove that as $n \rightarrow \infty$

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(a)

$\left\| \sigma_{n}-f\right\|_{\infty }=O\left(n^{-1} \ln n\right)$ ;

(b)

$\left\| S_{n}-f\right\|_{2}=O\left(n^{-1}\right)$ ;

(c)

$\left\| S_{n}-f\right\|_{\infty }=O\left(n^{-1} \ln n\right)$ .

Problem IX.4.22

Let $f$ be a $2 \pi$ -periodic function, $f \in \operatorname {Lip}_{\alpha }$ . Prove that if $\alpha >\frac{1}{2}$ , then $\sum_{n \in \mathbb {Z}}\left|\widehat{f}(n)\right|<\infty$ . For $\alpha =\frac{1}{2}$ the statement becomes false (see problem 4.30).

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Problem IX.4.23

The Cesàro-Fejér sum of a function $f \in \mathscr {L}^{1}(-\pi , \pi )$ is the trigonometric polynomial

\sigma _{n}(f, x)=\sigma _{n}(x)=\frac{S_{0}(x)+S_{1}(x)+\ldots +S_{n-1}(x)}{n} \quad (x \in \mathbb {R}),

\sigma _{n}(f, x)=\int _{-\pi }^{\pi } \Phi _{n}(x-y) f(y) d y, \quad \text{ where } \quad \Phi _{n}(x)=\frac{1}{2 \pi n} \frac{\sin ^{2}(n x / 2)}{\sin ^{2}(x / 2)};

moreover, $\sigma_{n}(f, x)=\sum_{\left|k\right|<n}\left(1-\frac{\left|k\right|}{n}\right) \widehat{f}(k) e^{i k x}$ .

Let $f$ be a $2 \pi$ -periodic function, summable on $(-\pi , \pi )$ . Prove the inequality

\sup _{n}\left|\sigma _{n}(f, x)\right| \leqslant C M(f, x)

where $C$ is an absolute constant, and $M(f, x)$ is the Hardy-Littlewood maximal function:

M(x)=\sup _{h>0} \frac{1}{2 h} \int _{x-h}^{x+h}\left|f(y)\right| d y

Show by an example that the function $x \mapsto \sup_{n}\left|\sigma_{n}(f, x)\right|$ may fail to be summable on the interval $(0,2 \pi )$ .

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Problem IX.4.24

(a)(b)(c)

Let $f(x)=\sum_{n \geqslant 2} \frac{\cos n x}{n^{2}} \ln n \quad (x \in \mathbb {R})$ . Prove that

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(a)

$f \in C^{1}(\mathbb {R}), f \in C^{2}(0,2 \pi ), f^{\prime \prime } \in \mathscr {L}^{1}(0,2 \pi )$ ;

(b)

$\left\| f-S_{n}\right\|_{\infty } \asymp \frac{1}{n \ln n}$ ;

(c)

$E_{n}(f) \asymp \frac{1}{n \ln n}$ . Here $E_{n}(f)=\inf \left\| f-T\right\|_{\infty }$ , where the infimum is taken over all trigonometric polynomials $T$ whose order does not exceed $n$ .

Problem IX.4.25

(a)(b)(c)

Consider two sequences of algebraic polynomials $\left\{ P_{n}\right\}_{n \geqslant 0}$ and $\left\{ Q_{n}\right\}_{n \geqslant 0}$ (the Rudin-Shapiro polynomials), which are defined recursively as follows: $P_{0}=Q_{0}=1$ and for $n=0,1,2, \ldots$

P_{n+1}(z)=P_{n}(z)+z^{2^{n}} Q_{n}(z), \quad Q_{n+1}(z)=P_{n}(z)-z^{2^{n}} Q_{n}(z)

Prove that

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(a)

the degree of the polynomials $P_{n}$ and $Q_{n}$ equals $2^{n}-1$ ;

(b)

all coefficients of the polynomials $P_{n}$ and $Q_{n}$ equal $\pm 1$ ;

(c)

if $z \in \mathbb {C},\left|z\right|=1$ , then $\left|P_{n}(z)\right|^{2}+\left|Q_{n}(z)\right|^{2}=2^{n+1}$ .

Problem IX.4.26

Let $R_{N}(N=0,1,2, \ldots )$ be the natural projection of the set of all algebraic polynomials onto the set of polynomials of degree not exceeding $N$ ( $R_{N}$ annihilates monomials whose degree is greater than $N$ , and leaves unchanged monomials whose degree does not exceed $N$ ).

Prove that for $z \in \mathbb {C},\left|z\right|=1$ , and any $N, n \in \mathbb {N}$ , for the Rudin-Shapiro polynomials $P_{n}$ and $Q_{n}$ (see problem 4.25) the following inequalities hold:

\left|R_{N}\left(P_{n}\right)(z)\right| \leqslant 10 \sqrt{N}, \quad \left|R_{N}\left(Q_{n}\right)(z)\right| \leqslant 10 \sqrt{N} .

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Problem IX.4.27

(a)(b)

Prove that there exists a series $\sum_{k \geqslant 0} \varepsilon_{k} e^{i k \theta }$ , where $\varepsilon_{k}= \pm 1$ , for all of whose partial sums $S_{n}$ the following inequalities hold:

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(a)

$\max_{0 \leqslant \theta \leqslant 2 \pi }\left|S_{n}(\theta )\right| \leqslant 10 \sqrt{n}$ ;

(b)

$\int_{0}^{2 \pi }\left|S_{n}(\theta )\right| d \theta \geqslant \sqrt{\frac{n}{3}}$ .

Problem IX.4.28

Using the sequence $\left\{ \varepsilon_{k}\right\}_{k \geqslant 0}$ from problem 4.27, verify that there exists a uniformly convergent series on $\mathbb {R}$ , $f(\theta )=\sum_{k \geqslant 0} a_{k} e^{i k \theta }$ , such that $\sum \left|a_{k}\right|^{p}=+\infty$ for any $p<2$ (Carleman's singularity).

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Problem IX.4.29

(a)(b)(c)

Let the sequence $\left\{ \varepsilon_{k}\right\} \subset \mathbb {C}$ satisfy the condition $\max_{\theta \in \mathbb {R}}\left|\sum_{k=1}^{n} \varepsilon_{k} e^{i k \theta }\right|=O(\sqrt{n})$ (cf. problem 4.27). Prove that for $\alpha >0$

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(a)

$\max_{\theta \in \mathbb {R}}\left|\sum_{1 \leqslant k \leqslant n!} \frac{\varepsilon_{k}}{k^{1 / 2-\alpha }} e^{i k \theta }\right|=O\left(n^{\alpha }\right)$ ;

(b)

$\max_{\theta \in \mathbb {R}}\left|\sum_{k \geqslant n} \frac{\varepsilon_{k}}{k^{1 / 2+\alpha }} e^{i k \theta }\right|=O\left(n^{-\alpha }\right)$ ;

(c)

the function $f(\theta )=\sum \frac{\varepsilon_{k}}{k^{1 / 2+\alpha }} e^{i k \theta }$ belongs to the class $\operatorname {Lip}_{\alpha }$ for $\alpha \in (0,1)$ .

Problem IX.4.30

Prove that there exists a function $f \in \operatorname {Lip}_{1 / 2}$ of the form $f(\theta )=\sum c_{k} e^{i k \theta }$ such that $\sum \left|c_{k}\right|=\infty$ (cf. problem 4.22).

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Problem IX.4.31

(a)(b)

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(a)

Let $f, g \in \mathscr {L}^{2}(-\pi , \pi ), h=f g$ . Prove that if $\widehat{f}(n)=\widehat{g}(n)=0$ for $n<0$ , then also $\widehat{h}(n)=0$ for $n<0$ .

(b)

Let $f \in \mathscr {L}^{\infty }(-\pi , \pi ), \Phi =e^{f}$ . Prove that if $\widehat{f}(n)=0$ for $n<0$ , then also $\widehat{\Phi }(n)=0$ for $n<0$ .

Problem IX.4.32

Let $f \in \mathscr {L}^{2}(-\pi , \pi )$ . Consider the function $\widetilde{f}$ , defined by the equality

\tilde{f}(x)=-i \sum _{n \in \mathbb {Z}} \widehat{f}(n) \operatorname {sign}(n) e^{i n x} \quad (x \in (-\pi , \pi )) .

Prove that the function $\widetilde{f}$ is real if the function $f$ is real.

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Problem IX.4.33

(a)(b)

Let $1=n_{0}<n_{1}<n_{2}<\ldots <n_{m}$ be natural numbers, and let $a_{1}$ , $a_{2}, \ldots , a_{m}$ be arbitrary complex numbers, and let

f_{k}(x)=\sum _{2^{k-1} \leqslant j<2^{k}} a_{j} e^{i n_{j} x} \quad (x \in \mathbb {R})

(we assume that $k \geqslant 1$ and $a_{j}=0$ for $j>m$ ). Further, let

g_{k}=\left|f_{k}\right|, \quad h_{k}=g_{k}-i \tilde{g}_{k}, \quad \text{ and } \quad F_{\varepsilon }=\sum _{k \geqslant 1} f_{k} e^{-\varepsilon \Sigma _{j \geqslant k} h_{j}},

where $\varepsilon$ is an arbitrary positive number, $\widetilde{g}_{k}$ is the function conjugate to $g_{k}$ (see the definition in problem 4.32). Prove that

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(a)

$\left|F_{\varepsilon }(x)\right| \leqslant \varepsilon^{-1}$ for any $x \in (-\pi , \pi )$ ;

(b)

if $2^{p-1} \leqslant q<2^{p}$ , then

\left|\widehat{F}_{\varepsilon }\left(n_{q}\right)-a_{q}\right| \leqslant \frac{\varepsilon }{2} \sum _{k \geqslant p}\left\| f_{k}\right\| _{2} \sum _{j \geqslant k}\left\| f_{j}\right\| _{2} .

Remark.

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The function $\widetilde{f}$ defined in problem 4.32 is called conjugate to $f$ . The reader can become acquainted with one of the numerous applications of this important notion in problems 4.33 and 4.34, borrowed from [MGPS].

Problem IX.4.34

Let $1 \leqslant n_{1}<n_{2}<\ldots <n_{m}\left(n_{j} \in \mathbb {N}\right)$ . Using the result of the previous problem, prove that for any complex numbers $c_{1}, c_{2}, \ldots , c_{m}$ the inequality

\int _{-\pi }^{\pi }\left|\sum _{1 \leqslant j \leqslant m} c_{j} e^{i n_{j} x}\right| d x \geqslant \delta \sum _{1 \leqslant j \leqslant m} \frac{\left|c_{j}\right|}{j},

holds, where $\delta$ is a positive absolute constant ( $\delta \geqslant 10^{-2}$ ).

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Remark.

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Problem IX.4.35

(a)(b)(c)

The Fourier transform of a function $f \in \mathscr {L}^{1}\left(\mathbb {R}^{m}\right)$ is the function $\widehat{f}$ , defined by the equality

\widehat{f}(s)=\int _{\mathbb {R}^{m}} f(t) e^{-i(s, t)} d t \quad \left(s \in \mathbb {R}^{m}\right),

where $(s, t)$ is the scalar product of the vectors $s, t \in \mathbb {R}^{m}$ . The convolution of two summable functions $g$ and $h$ on $\mathbb {R}^{m}$ is denoted by $g * h$ : by definition $(g * h)(x)=\int_{\mathbb {R}^{m}} g(x-y) h(y) d y$ for almost all $x \in \mathbb {R}^{m}$ .

Find the Fourier transform and the convolution $f_{c_{1}} * f_{c_{2}}$ of the functions in the following cases ( $c>0$ ):

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(a)

$f_{c}(x)=\frac{1}{c \sqrt{2 \pi }} e^{-x^{2} /\left(2 c^{2}\right)} \quad (x \in \mathbb {R})$ ;

(b)

$f_{c}(x)=\left(2 \pi c^{2}\right)^{-m / 2} e^{-\left\| x\right\|^{2} /\left(2 c^{2}\right)}\left(x \in \mathbb {R}^{m}\right)$ ;

(c)

$f_{c}(x)=\frac{1}{\pi } \frac{c}{c^{2}+x^{2}} \quad (x \in \mathbb {R})$ .

Problem IX.4.36

The Fourier transform of a function $f \in \mathscr {L}^{1}\left(\mathbb {R}^{m}\right)$ is the function $\widehat{f}$ defined by $\widehat{f}(s)=\int_{\mathbb {R}^{m}} f(t) e^{-i(s, t)} d t$ ( $s \in \mathbb {R}^{m}$ ), where $(s, t)$ is the scalar product of the vectors $s, t \in \mathbb {R}^{m}$ . Find the Fourier transforms of the Hermite functions

h_{n}(x)=e^{x^{2} / 2}\left(e^{-x^{2}}\right)^{(n)} \quad (x \in \mathbb {R}, n=0,1,2, \ldots ) .

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Problem IX.4.37

The convolution of two summable functions $g$ and $h$ on $\mathbb {R}^{m}$ is denoted by $g * h$ : by definition $(g * h)(x)=\int_{\mathbb {R}^{m}} g(x-y) h(y) d y$ for almost all $x \in \mathbb {R}^{m}$ . The Fourier transform of a function $f \in \mathscr {L}^{1}\left(\mathbb {R}^{m}\right)$ is the function $\widehat{f}(s)=\int_{\mathbb {R}^{m}} f(t) e^{-i(s, t)} d t$ ( $s \in \mathbb {R}^{m}$ ); it is injective in the sense that if $\widehat{f}=\widehat{g}$ , then $f(t)=g(t)$ for almost all $t \in \mathbb {R}^{m}$ .

Prove that if the convolution of a function $f$ summable on $\mathbb {R}^{m}$ with the function $e^{-\left\| x\right\|^{2}}$ is identically zero, then $f(x)=0$ almost everywhere.

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Problem IX.4.38

The Fourier transform of a finite measure $\mu$ , defined at least on the Borel subsets of the space $\mathbb {R}^{m}$ , is the function $\widehat{\mu }$ defined by the equality

\widehat{\mu }(s)=\int _{\mathbb {R}^{m}} e^{-i(s, t)} d \mu (t) \quad \left(s \in \mathbb {R}^{m}\right),

where $(s, t)$ is the inner product of the vectors $s, t \in \mathbb {R}^{m}$ . Like the Fourier transform of summable functions, the Fourier transform of measures is injective (see, for example, p. 302 in [Shi]). The Fourier transform of the measure on $\mathbb {R}$ generated by a nondecreasing bounded function $g$ we denote by the symbol $\widehat{g}$ .

Let $F$ be the increasing distribution function of the function $f \in \mathscr {L}^{0}(E)$ , where $E \subset \mathbb {R}^{m}, \lambda_{m}(E)<\infty$ . Prove that

\widehat{F}(s)=\int _{E} e^{-i s f(x)} d x \quad (s \in \mathbb {R})

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Problem IX.4.39

Let $\widehat{\sigma }(s)=\int_{0}^{1} e^{-i s t} d \sigma (t)(s \in \mathbb {R})$ , where $\sigma$ is the Cantor function (see the definition before problem III.3.14). Prove that $\widehat{\sigma }(s) \nrightarrow 0$ as $s \rightarrow \infty$ .

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Problem IX.4.40

(a)(b)

The Fourier transform of the measure on $\mathbb {R}$ generated by a nondecreasing bounded function $g$ is denoted by $\widehat{g}$ : $\widehat{g}(s)=\int_{\mathbb {R}} e^{-i s t} d g(t)$ ( $s \in \mathbb {R}$ ). Let $\sigma_{1}(x)=\int_{0}^{1} \sigma (x-y) d \sigma (y)(x \in \mathbb {R})$ , where $\sigma$ is the Cantor function ( $\sigma (x)=0$ for $x<0, \sigma (x)=1$ for $x>1$ ). Prove that

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(a)

$\sigma_{1}$ is strictly increasing on the interval $\left[0,2\right]$ ;

(b)

$\widehat{\sigma }_{1}(s) \nrightarrow 0$ as $s \rightarrow \infty$ .

Problem IX.4.41

(a)(b)(c)(d)

g(t)=\sup \left\{ \sum _{k \geqslant 1} \varepsilon _{k} 2^{-k} \mid \sum _{k \geqslant 1} \varepsilon _{k} 2^{-(2 k-1)} \leqslant t, \quad \varepsilon _{k}=0 \text{ or } 1\right\} ,

$g(t)=0$ for $t<0$ . Prove that

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(a)

$g \in C(\mathbb {R})$ ;

(b)

$g^{\prime }(t)=0$ almost everywhere on $\mathbb {R}$ ;

(c)

$\widehat{g}(s)=e^{-i s / 3} \prod_{n \geqslant 1} \cos \left(s / 4^{n}\right)$ ;

(d)

$\quad \int_{-\infty }^{+\infty } g(x-t) d g(2 t)= \begin{cases} 0 & \text{ for } x<0, \\ x & \text{ for } 0 \leqslant x \leqslant 1, \\ 1 & \text{ for } x>1 . \end{cases}$

Problem IX.4.42

(a)(b)(c)

Let $v=\left\{ n_{k}\right\}$ be an arbitrary strictly increasing sequence of natural numbers, $\nu_{1}=\left\{ n_{2 k-1}\right\}_{k \geqslant 1}, \nu_{2}=\left\{ n_{2 k}\right\}_{k \geqslant 1}$ . Further, let, for $t>0$ ,

g(t)=\sup \left\{ \sum _{k \geqslant 1} \varepsilon _{k} 2^{-k} \mid \sum _{k \geqslant 1} \varepsilon _{k} 2^{-n_{k}} \leqslant t, \quad \varepsilon _{k}=0 \text{ or } 1\right\} ,

$g(t)=0$ for $t \leqslant 0$ , and let the functions $g_{1}$ and $g_{2}$ be constructed analogously using the sequences $v_{1}$ and $v_{2}$ respectively. Prove that

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(a)

$g \in C(\mathbb {R})$ ;

(b)

$g_{1}^{\prime }(t)=g_{2}^{\prime }(t)=0$ almost everywhere on $\mathbb {R}$ ;

(c)

the function $g$ is "divisible" in the sense that $g=g_{1} * g_{2}$ , that is,

g(x)=\int _{-\infty }^{+\infty } g_{1}(x-t) d g_{2}(t) \text{ for any } x \in \mathbb {R} \text{. }

Problem IX.4.43

Prove that if two finite Borel measures in $\mathbb {R}^{m}$ take equal values on all possible half-spaces (i.e., sets of the form $\left\{ x \in \mathbb {R}^{m} \mid (x, a) \geqslant c\right\}$ , where $a \in \mathbb {R}^{m}, c \in \mathbb {R}$ ), then they coincide.

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Problem IX.4.44

Find the general form of a probability Borel measure in the space $\mathbb {R}^{m}$ that is invariant under orthogonal transformations and is at the same time a product of two measures concentrated on nontrivial orthogonal subspaces.

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Problem IX.4.45

Let $W$ be a simplex in $\mathbb {R}^{m}$ with vertices $v_{0}, \ldots , v_{m}$ . Prove that the Fourier transform of the characteristic function $\chi_{w}$ of the set $W$ equals

\widehat{\chi }_{W}(s)=\int _{W} e^{-i(s, t)} d t=\frac{m!\lambda _{m}(W)}{i^{m}} \sum _{j=0}^{m} e^{-i\left(s, v_{j}\right)} \prod _{\substack {k=0 \\ k \neq j}}^{m} \frac{1}{\left(s, v_{k}-v_{j}\right)}

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Problem IX.4.46

Let $\widehat{\chi }_{Q}(s)=\int_{Q} e^{-i(s, t)} d t$ be the Fourier transform of the characteristic function of the square $Q=\left[-1,1\right]^{2}$ . Since the function $\chi_{Q}$ is discontinuous, its Fourier transform is not summable on $\mathbb {R}^{2}$ . Prove that nevertheless $\widehat{\chi }_{Q} \in \mathscr {L}^{p}\left(\mathbb {R}^{2}\right)$ for any $p>1$ . What is the asymptotics of the integral $I_{R}=\int_{\left\| s\right\| \leqslant R}\left|\widehat{\chi }_{Q}(s)\right| d s$ as $R \rightarrow +\infty$ ?

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Problem IX.4.47

(a)(b)

Let $\widehat{\chi }_{B}(s)=\int_{B} e^{-i(s, t)} d t$ be the Fourier transform of the characteristic function of the disk $B=\left\{ t \in \mathbb {R}^{2} \mid \left\| t\right\| <1\right\}$ .

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(a)

For which $p>1$ does the inclusion $\widehat{\chi }_{B} \in \mathscr {L}^{p}\left(\mathbb {R}^{2}\right)$ hold?

(b)

What is the asymptotics of the integral $\mathcal{I}_{R}=\int_{\left\| s\right\| \leqslant R}\left|\widehat{\chi }_{B}(s)\right| d s$ as $R \rightarrow +\infty$ ?

Problem IX.4.48

A trigonometric polynomial in two variables $x, y \in \mathbb {R}$ is a sum of the form

T(x, y)=\sum _{m, n \in \mathbb {Z}} c_{m, n} e^{i(m x+n y)},

in which only finitely many of the coefficients $c_{m, n} \in \mathbb {C}$ are nonzero. With each set $\Omega \subset \mathbb {R}^{2}$ we associate the projector $P_{\Omega }$ , acting on the space of all such polynomials:

P_{\Omega }(T ; x, y)=\sum _{(m, n) \in \Omega } c_{m, n} e^{i(m x+n y)}

(for brevity we will omit the indication $m, n \in \mathbb {Z}$ ). Thus the projector $P_{\Omega }$ retains in the trigonometric polynomial only those terms corresponding to frequencies in $\Omega$ . Allowing ourselves a slight abuse of language, we will say that $P_{\Omega }$ is "the projector onto $\Omega$ ". We call the projector $P_{\Omega }$ bounded if there exists a number $C$ such that for every polynomial $T$ the inequality

\max _{x, y \in \mathbb {R}}\left|P_{\Omega }(T ; x, y)\right| \leqslant C \max _{x, y \in \mathbb {R}}\left|T(x, y)\right|

holds. The smallest of all such numbers $C$ we denote by $\left\| P_{\Omega }\right\|$ .

The projector and its norm are defined analogously in the space of trigonometric polynomials of one variable. The projector onto the set $\Omega =\left[-N, N\right]$ is none other than the $N$ -th partial sum of the Fourier series. It is clear that if the set $\Omega \subset \mathbb {R}$ is bounded, then

P_{\Omega }(T ; x)=\int _{-\pi }^{\pi } T(u) D_{\Omega }(x-u) d u,

where $D_{\Omega }(u)=\frac{1}{2 \pi } \sum_{n \in \Omega } e^{-i n u}$ is the Dirichlet kernel corresponding to the set $\Omega$ . In this case the projector $P_{\Omega }$ is naturally defined not only on trigonometric polynomials, but also on all functions summable on $\left[-\pi , \pi \right]$ . One can prove that

\left\| P_{\Omega }\right\| =\left\| D_{\Omega }\right\| _{1}=\int _{-\pi }^{\pi }\left|D_{\Omega }(u)\right| d u<\infty .

For $\Omega_{N}=\left[-N, N\right]$ the function $D_{\Omega_{N}}(u)=\frac{1}{2 \pi } \sum_{\left|n\right| \leqslant N} e^{-i n u}$ is the classical Dirichlet kernel and

\left\| P_{\Omega _{N}}\right\| =\int _{-\pi }^{\pi }\left|D_{\Omega _{N}}(u)\right| d u=\frac{1}{2 \pi } \int _{-\pi }^{\pi }\left|\frac{\sin \left(N+\frac{1}{2}\right) u}{\sin \frac{u}{2}}\right| d u .

Note that $\left\| P_{\Omega_{N}}\right\| \asymp \ln N$ as $N \rightarrow \infty$ (see problem IV.6.11). Analogous facts hold for polynomials in two variables as well. If the set $\Omega \subset \mathbb {R}^{2}$ is bounded, then

P_{\Omega }(T ; x, y)=\iint _{\left[-\pi , \pi \right]^{2}} T(u, v) D_{\Omega }(x-u, y-v) d u d v,

where $D_{\Omega }(u, v)=\frac{1}{4 \pi^{2}} \sum_{(m, n) \in \Omega } e^{-i(m u+n v)}$ is the "two-dimensional" Dirichlet kernel corresponding to the set $\Omega$ . As in the one-dimensional case,

\left\| P_{\Omega }\right\| =\left\| D_{\Omega }\right\| _{1}=\iint _{\left[-\pi , \pi \right]^{2}}\left|D_{\Omega }(u, v)\right| d u d v<\infty .

Prove that the projector onto the ray $\Omega =(a,+\infty )$ is not bounded.

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Problem IX.4.49

For a set $\Omega \subset \mathbb {R}$ the projector $P_{\Omega }$ retains in a trigonometric polynomial $T(x)=\sum_{n} c_{n} e^{i n x}$ only the terms with frequencies $n \in \Omega$ : $P_{\Omega }(T ; x)=\sum_{n \in \Omega } c_{n} e^{i n x}$ (similarly for polynomials $T(x, y)=\sum c_{m, n} e^{i(m x+n y)}$ in two variables and $\Omega \subset \mathbb {R}^{2}$ ). The projector is called bounded if $\max \left|P_{\Omega }(T)\right| \leqslant C \max \left|T\right|$ for all $T$ ; the smallest such $C$ is denoted $\left\| P_{\Omega }\right\|$ . For bounded $\Omega$ we have $\left\| P_{\Omega }\right\| =\left\| D_{\Omega }\right\|_{1}$ , where $D_{\Omega }(u)=\frac{1}{2 \pi } \sum_{n \in \Omega } e^{-i n u}$ is the Dirichlet kernel (in the two-dimensional case $D_{\Omega }(u, v)=\frac{1}{4 \pi^{2}} \sum_{(m, n) \in \Omega } e^{-i(m u+n v)}$ , with the norm in $\mathscr {L}^{1}\left(\left[-\pi , \pi \right]^{2}\right)$ ).

Let $N \in \mathbb {N}$ . Prove that

\frac{1}{2 \pi } \int _{-\pi }^{\pi }\left|\frac{\sin \left(N+\frac{1}{2}\right) u}{\sin \frac{u}{2}}\right| d u=\frac{16}{\pi ^{2}} \sum _{k \geqslant 1} \frac{1+\frac{1}{3}+\ldots +\frac{1}{2 k(2 N+1)-1}}{4 k^{2}-1} .

In particular, the norms of the projectors onto $\left[-N, N\right]$ increase as $N$ grows.

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Problem IX.4.50

Let $\Omega_{0} \subset \Omega \subset \mathbb {R}$ . Is it true that $\left\| P_{\Omega_{0}}\right\| \leqslant \left\| P_{\Omega }\right\|$ , if $\Omega$ is a bounded set?

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Problem IX.4.51

Using the result of problem 4.34, prove that the norms of the projectors onto the intervals $\Omega_{N}=\left[-N, N\right]$ have the minimal order of growth: for any set $\Omega \subset \mathbb {R}, \operatorname {card}(\Omega \cap \mathbb {Z})=N$ , the inequality $\left\| P_{\Omega }\right\| \geqslant c\left\| P_{\Omega_{N}}\right\|$ holds with an absolute constant $c>0$ .

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Problem IX.4.52

Prove that for any half-plane $\Omega \subset \mathbb {R}^{2}$ the projector $P_{\Omega }$ is not bounded. Is the projector onto the strip

\Omega _{\Delta }=\left\{ (s, t) \in \mathbb {R}^{2} \mid \left|t\right| \leqslant \Delta \right\} ?

bounded?

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Problem IX.4.53

Prove that the projector onto the inclined strip

\Omega =\left\{ (s, t) \in \mathbb {R}^{2} \mid \left|t-a s\right| \leqslant \Delta \right\} \quad (a \in \mathbb {R}, \Delta >0)

with slope coefficient $a \in \mathbb {R}$ is bounded only for rational $a$ .

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Problem IX.4.54

Let $\Omega \subset \mathbb {R}^{2}$ and $N=\operatorname {card}\left(\Omega \cap \mathbb {Z}^{2}\right)<+\infty$ . Prove that the trivial estimate

\left\| P_{\Omega }\right\| \leqslant 2 \pi \left\| D_{\Omega }\right\| _{2}=\sqrt{N},

cannot in general be improved in order: there exists a subset $\Omega_{0} \subset \Omega$ such that $\left\| P_{\Omega_{0}}\right\| \geqslant c \sqrt{N}$ , where $c>0$ is an absolute constant.

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Problem IX.4.55

Prove that the norm of the projector onto the square $\left[-R, R\right]^{2}$ has growth order $\ln^{2} R$ (as $R \rightarrow +\infty$ ), while the norms of the projectors onto the disk $B(R)$ grow significantly faster - like $\sqrt{R}$ .

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Problem IX.4.56

(a)(b)(c)

Let $S_{N, M}$ be the projectors onto the rectangles $\Omega_{N, M}= =\left[-N, N\right] \times [-M, M](N, M \in \mathbb {N})$ . Consider $2 \pi$ -periodic functions $f_{\nu }(\nu \geqslant 1)$ in both variables such that $f_{\nu }(x, y)=e^{i \nu x y}$ for $x, y \in (0,2 \pi )$ . Prove that for any $x$ and $y$ from $(0,2 \pi )$ the following inequalities hold:

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(a)

$\left|S_{N, M}\left(f_{v} ; x, y\right)\right| \leqslant \frac{4}{\pi } \ln N+O(1)$ ;

(b)

if $M>2 \pi \nu$ , then $\left|S_{N, M}\left(f_{\nu } ; x, y\right)\right| \leqslant O\left(\frac{\ln N}{M-2 \pi \nu }\right)+O(1)$ ;

(c)

$\left|S_{N, M}\left(f_{\nu } ; x, y\right)\right| \geqslant \frac{\ln \nu }{2 \pi }+O(1)$ for $N=\left[\nu y\right]$ and $M=\left[\nu x\right]$ (the constants in the $O$ -terms depend only on $x$ and $y$ ).

Problem IX.4.57

Using the result of the previous problem, construct a bounded and continuous function $F$ on $(0,2 \pi )^{2}$ for which

\sup _{N, M}\left|S_{N, M}(F ; x, y)\right|=\infty \quad \text{ at every point } \quad (x, y) \in (0,2 \pi )^{2}

(Fefferman's example). Thus, the rectangular partial sums of the double Fourier series of the function $F$ diverge unboundedly everywhere on $(0,2 \pi )^{2}$ .

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