Conjugate function and Hilbert transform

Prove that the system $\left\{ e^{i n t}\right\}_{-\infty }^{\infty }$ is a Schauder basis in the spaces $L_{p}[0,2 \pi ], p \in (1, \infty )$.

Find the spectrum of the Hilbert operator in $L_{2}(\mathbb {R})$:

The Hilbert transform of a function $x \in L_{2}[0,2 \pi ]$ is the function $y(t) = \frac{1}{2 \pi } \int_{0}^{2 \pi } x(s) \operatorname {ctg} \frac{t-s}{2} d s.{ }$¹ Prove that the operator $A: x \mapsto y$ is a partial isometry.

To each real function $u$ in $L^{2}$ there corresponds a unique real function $v$ in $L^{2}$ such that $\left\langle v, e_{0} \right\rangle = 0$ and such that $u + iv \in H^{2}$. Equivalently, to each real $u$ in $L^{2}$ there corresponds a unique $f$ in $H^{2}$ such that $\left\langle f, e_{0} \right\rangle$ is real and such that $\mathrm{Re}\left(f\right) = u$.

The Hilbert transform of $f \in L^{2}(\mathbb {R})$ is the function $H f$ whose Fourier transform is $(H f)^{\wedge }(\xi ) = -i \operatorname {sign}\left(\xi \right) \widehat{f}(\xi )$, where $\operatorname {sign}\left(\xi \right) = 1,0$, or -1 according to whether $\xi > 0, \xi = 0$, or $\xi < 0$, respectively.

Prove Theorem 14.6.

Finish the proof of Theorem 14.7.

Given $1 < p < \infty$, show that conjugation is bounded on $L^{p}(\mathbf{T})$ if and only if it is bounded on $L^{p^{\prime }}(\mathbf{T})$.

The function $\bar{D}_{n}(x) = \sin x+\sin 2 x+\ldots +\sin nx$ is called the conjugate Dirichlet kernel. Prove that:

The series conjugate to the trigonometric series (1) is called the series

Let series (1) be the Fourier series of the function $f$. Prove that in this case the partial sums $\bar{S}_{n}(x ; f)$ of the conjugate series can be written in the form

where $\bar{D}_{n}(x)$ is defined in problem 104.

Prove that if $f$ is a function Riemann-integrable on the segment $[-\pi ; \pi ]$, and hence there exists a constant $M > 0$ such that for all $x \in [-\pi ; \pi ]$ the inequality $\left|f(x)\right| \leqslant M$ holds, then

$\left|x\right| \leqslant \pi$, where $c$ is some absolute constant.

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

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

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 ($\delta \geqslant 10^{-2}$).

Put $c = (2 / \pi )^{1 / 2}$. For $j = 1,2,3, \ldots$, define $g_{j}$ on the real line by

Prove that $\left\{ \hat{g}_{j}\right\}$ is a uniformly bounded sequence of functions which converges pointwise, as $j \rightarrow \infty$. If $f \in L^{2}\left(\mathbb {R}^{1}\right)$, it follows that $f * g_{j}$ converges, in the $L^{2}$ metric, to a function $H f \in L^{2}$. This is the Hilbert transform of $f$; formally,

(The integral exists, in the principal value sense, for almost every $x$, but this is not so easy to prove; if $f$ satisfies a Lipschitz condition of order 1, for instance, the proof is trivial). Prove that

for every $f \in L^{2}\left(\mathbb {R}^{1}\right)$. Thus $H$ is an $L^{2}$-isometry, of period 4. Is it true that $H f \in \mathscr {S}_{1}$ if $f \in \mathscr {S}_{1}$?

For $f \in L^{2}(R), x \in R, 0 < y < \infty$, define

and put $Q(0) f = f$. Show that $\left\{ Q(y): 0 \leq y < \infty \right\}$ satisfies the conditions imposed in Definition 13.34 and that $\left\| Q(y)\right\| = 1$ for all $y$.

Find the domain of the infinitesimal generator $A$ of $\left\{ Q(y)\right\}$, and prove that

where $H$ is the Hilbert transform (Chapter 7, Exercise 24). Prove that $-A$ is positive and self-adjoint.