Rademacher Functions. The Khinchin Inequality

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?

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

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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

where $A, B$ are absolute constants.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $\lambda (E) \leqslant \frac{A}{t}$, $\lambda (E) \leqslant 2 e^{-t^{2} /\left(2 A^{2}\right)}$.

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

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 $s>0$.

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

Sharpen this inequality for the Rademacher system.

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

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 $n \geqslant m$

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

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

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 $\Delta_{n, k}=\left(\frac{k}{2^{n}}, \frac{k+1}{2^{n}}\right), k=0,1, \ldots , 2^{n}-1$;

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

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

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 $\lambda (E) \leqslant \frac{\left\| f\right\|_{1}}{t}$.

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:

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).

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 $m \in \mathbb {N}$.

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

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

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

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

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

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

where $c>0$ is an absolute constant.

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

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 $n)$.