Inequalities for trigonometric polynomials

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

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

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 $\Phi_{n}$:

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

Deduce from this the weakened Bernstein inequality:

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

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 $n=0,1,2, \ldots$

Prove that

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:

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:

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).

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 $\alpha >0$

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

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

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

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

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

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 $\mathscr {L}^{1}\left(\left[-\pi , \pi \right]^{2}\right)$).

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 $c>0$.