The Integral Transform of N.I.Akhiezer

TL;DR

This paper investigates a specific integral transform introduced by N.I. Akhiezer, analyzing its properties and applications as presented in his textbook on integral transforms.

Contribution

It provides a detailed study of Akhiezer's integral transform, clarifying its form, properties, and potential uses in mathematical analysis.

Findings

Characterization of the transform's properties

Connections to classical integral transforms

Potential applications in solving integral equations

Abstract

We study the integral transform which appeared in a different form in Akhiezer's textbook "Lectures on Integral Transforms".

Equations

$$({\mathbf{\Phi }}_{\omega }\mathbf{x})(t)=\underset{\mathbb{R}}{\int }{\Phi }_{\omega }(t-\tau )\mathbf{x}(\tau )d\tau ,t\in \mathbb{R},$$

$$({\mathbf{\Psi }}_{\omega }\mathbf{x})(t)=\underset{\mathbb{R}}{\int }{\Psi }_{\omega }(t-\tau )\mathbf{x}(\tau )d\tau ,t\in \mathbb{R}.$$

$$\mathbf{x}(\tau )=\left[\begin{array}{c}{x}_1(\tau )\\ x_2(\tau )\end{array}\right],$$

$${\Phi }_{\omega }(t)=\left[\begin{array}{cc}{C}_{\omega }(t)& S_{\omega }(t)\\ S_{\omega }(t)& C_{\omega }(t)\end{array}\right],$$

$${\Psi }_{\omega }(t)=\left[\begin{array}{cc}{C}_{\omega }(t)& -S_{\omega }(t)\\ -S_{\omega }(t)& C_{\omega }(t)\end{array}\right],$$

$$C_{\omega }(t)=\frac{\omega }{\pi }\cdot \frac{1}{\cosh \omega t},S_{\omega }(t)=\frac{\omega }{\pi }\cdot \frac{1}{\sinh \omega t}\cdot$$

$$\sinh z=\frac{e^z-e^{-z}}{2},\cosh z=\frac{e^z+e^{-z}}{2},\tanh z=\frac{e^z-e^{-z}}{e^z+e^{-z}},\mathrm{sech}z=\frac{2}{e^z+e^{-z}}\cdot$$

$${\mathbf{\Phi }}_{\omega }=\left[\begin{array}{cc}{\mathbf{C}}_{\omega }& {\mathbf{S}}_{\omega }\\ {\mathbf{S}}_{\omega }& {\mathbf{C}}_{\omega }\end{array}\right],{\mathbf{\Psi }}_{\omega }=\left[\begin{array}{cc}{\mathbf{C}}_{\omega }& -{\mathbf{S}}_{\omega }\\ -{\mathbf{S}}_{\omega }& {\mathbf{C}}_{\omega }\end{array}\right],$$

$$({\mathbf{C}}_{\omega }x)(t)=\underset{\mathbb{R}}{\int }{C}_{\omega }(t-\tau )x(\tau )d\tau ,$$

$$({\mathbf{S}}_{\omega }x)(t)=\underset{\mathbb{R}}{\int }{S}_{\omega }(t-\tau )x(\tau )d\tau .$$

$$\frac{\omega }{\pi }{e}^{-\omega |\xi |}\le C(\xi )<\frac{2\omega }{\pi }{e}^{-\omega |\xi |},\forall \xi \in \mathbb{R}.$$

$$|\tau |-|t|\le |t-\tau |\le |\tau |+|t|,$$

$$\frac{\omega }{\pi }{e}^{-\omega |t|}{e}^{-\omega |\tau |}\le C(t-\tau )<\frac{2\omega }{\pi }{e}^{\omega |t|}{e}^{-\omega |\tau |},\forall t\in \mathbb{R},\forall \tau \in \mathbb{R}.$$

$$\underset{\mathbb{R}}{\int }|x(\xi )|e^{-\omega |\xi |}d\xi <\infty .$$

$$|S(\xi )|<\frac{2\omega }{\pi (1-e^{-2\omega |\xi |})}{e}^{-\omega |\xi |},\forall \xi \in \mathbb{R}.$$

$$S_{\omega }(\xi )=\frac{1}{\pi \xi }+r(\xi ),$$

$$\underset{(t-\epsilon ,t+\epsilon )}{\int }|S_{\omega }(t-\tau )|d\tau =\infty ,\forall t\in \mathbb{R},\forall \epsilon >0.$$

$$\underset{\mathbb{R}}{\int }|S_{\omega }(t-\tau )x(\tau )|d\tau =\infty$$

$$p.v.\underset{\mathbb{R}}{\int }{S}_{\omega }(t-\tau )x(\tau )d\tau \stackrel{\text{def}}{=}\underset{\epsilon \to +0}{\lim }\underset{\mathbb{R}\setminus (t-\epsilon ,t+\epsilon )}{\int }{S}_{\omega }(t-\tau )x(\tau )d\tau$$

$$\underset{\mathbb{R}\setminus (t-\epsilon ,t+\epsilon )}{\int }|S_{\omega }(t-\tau )x(\tau )|d\tau <\infty ,\forall t\in \mathbb{R},\forall \epsilon >0.$$

$$y(t)=\text{p.v.}\underset{\mathbb{R}}{\int }{S}_{\omega }(t-\tau )x(\tau )d\tau ,$$

$$({\mathbf{\Phi }}_{\omega }\mathbf{x})(t)=\mathbf{y}(t),({\mathbf{\Psi }}_{\omega }\mathbf{x})(t)=\mathbf{z}(t),$$

$$\mathbf{y}(t)=\left[\begin{array}{c}{y}_1(t)\\ y_2(t)\end{array}\right],\mathbf{z}(t)=\left[\begin{array}{c}{z}_1(t)\\ z_2(t)\end{array}\right],$$

$$y_1(t)=({\mathbf{C}}_{\omega }{x}_1)(t)+({\mathbf{S}}_{\omega }{x}_2)(t),$$

$$y_2(t)=({\mathbf{S}}_{\omega }{x}_1)(t)+({\mathbf{C}}_{\omega }{x}_2)(t),$$

$$\Vert u{\Vert }_{L^1}=\underset{\mathbb{R}}{\int }|u(t)|dt.$$

$$\|u\|_{L^{2}}=\Big{\{}\int\limits_{\mathbb{R}}|u(\xi)|^{2}d\xi\Big{\}}^{1/2}.$$

$$⟨u^{\prime },u^{\prime \prime }{⟩}_{L^2}=\underset{\mathbb{R}}{\int }{u}^{\prime }(t)\overline{u^{\prime \prime }(t)}dt.$$

$$\mathfrak{F}u=\hat{u}$$

$$\hat{u}(\lambda )=\underset{\mathbb{R}}{\int }u(t)e^{it\lambda }dt,$$

$$\Vert \hat{u}{\Vert }_{L^2}^2=2\pi \Vert u{\Vert }_{L^2}^2,\forall u\in L^2.$$

$${\mathfrak{F}}^{-1}v=\check{v},$$

Full text

1. The Akhiezer Integral Transforms: a formal definition.

In the present paper we consider the one-parametric family of pairs $\boldsymbol{\Phi}_{\omega},\boldsymbol{\Psi}_{\omega}$ of linear integral operators. The parameter $\omega$ which enumerates the family can be an arbitrary positive number and is fixed in the course of our consideration. Formally the operators $\boldsymbol{\Phi}_{\omega},\boldsymbol{\Psi}_{\omega}$ are defined as convolution operators according the formulas

[TABLE]

In (1.1), $\boldsymbol{x}(\tau)$ is $2\times 1$ vector-column,

[TABLE]

which entries $x_{1}(\tau),x_{2}(\tau)$ are measurable functions, and $\varPhi_{\omega}$, $\varPsi_{\omega}$ are $2\times 2$ matrices,

[TABLE]

where

[TABLE]

Here and in what follows, $\sinh,\,\cosh,\tanh,\operatorname{sech}$ are hyperbolic functions.

For $z\in\mathbb{C}$

[TABLE]

The operators $\boldsymbol{\Phi}_{\omega},\,\boldsymbol{\Psi}_{\omega}$ are naturally decomposed into blocks:

[TABLE]

where $\boldsymbol{C}_{\omega}$ and $\boldsymbol{S}_{\omega}$ are convolution operators:

[TABLE]

In (1.6), $x$ is a $\mathbb{C}$-valued function.

The function $C_{\omega}(\xi)$ is continuous and positive on $\mathbb{R}$. It decays exponentially as $|\xi|\to\infty$:

[TABLE]

Since

[TABLE]

the convolution kernel $C(t-\tau)$ admits the estimate

[TABLE]

Definition 1.1**.**

The set $L_{\omega}^{1}$ as the set of all complex valued functions $x(t)$ which are measurable, defined almost everywhere with respect to the Lebesgue measure on $\mathbb{R}$ and satisfy the condition

[TABLE]

The set $L_{\omega}^{1}\dotplus{}L_{\omega}^{1}$ is the set of all $2\times 1$ columns $\boldsymbol{x}(t)=\begin{bmatrix}x_{1}(t)\\ x_{2}(t)\end{bmatrix}$ such that $x_{1}(t)\in{}L_{\omega}^{1}$ and $x_{2}(t)\in{}L_{\omega}^{1}$.

Lemma 1.2**.**

Let $x(\tau)$ be a $\mathbb{C}$-valued function which belongs to the space $L_{\omega}^{1}$. Then the integral in the right hand side of (1.6a) exists111That is the value of this integral is a finite complex number for every $t\in\mathbb{R}$. for every $t\in\mathbb{R}$.

We define the function $(\boldsymbol{C}_{\omega}x)(t)$ by means of the equality (1.6a).

Remark 1.3*.*

For $x(\tau)\in{}L^{1}_{\omega}$, the function $(\boldsymbol{C}_{\omega}x)(t)$ is a continuous function well defined on the whole $\mathbb{R}$. Nevertheless the function $(\boldsymbol{C}_{\omega}x)(t)$ may not belong to the space $L^{1}_{\omega}$. The operator $\boldsymbol{C}_{\omega}$ does not map the space $L^{1}_{\omega}$ into itself. (In other words, the operator $\boldsymbol{C}_{\omega}$ considered as an operator in $L^{1}_{\omega}$ is unbounded.)

The situation with the integral in the right hand side of (1.6b) is more complicated. The function $S_{\omega}(\xi)$ also decays exponentially as $|\xi|\to\infty$:

[TABLE]

However the function $S_{\omega}$ has the singularity at the point $\xi=0$:

[TABLE]

where $r(\xi)$ is a function which is continuous and bounded for $\xi\in\mathbb{R}$. Thus the convolution kernel $S_{\omega}(t-\tau)$ has a non-integrable singularity on the diagonal $t=\tau$:

[TABLE]

Therefore the integral in the right hand side of (1.6b) may not exist as a Lebesgue integral. Given a function $x(\tau)$, the equality

[TABLE]

holds at every point $t\in\mathbb{R}$ which is a Lebesgue point of the function $x$ and $x(t)\not=0$. Nevertheless, under the condition (1.9) we can attach a meaning to the integral $\int\limits_{\mathbb{R}}S_{\omega}(t-\tau)x(\tau)d\tau$ for almost every $t\in\mathbb{R}$.

Lemma 1.4**.**

Let $x(\tau)$ be a $\mathbb{C}$-valued function which belongs to the space $L_{\omega}^{1}$. Then the principal value integral

[TABLE]

exists for almost every $t\in\mathbb{R}$.

We define the function $(\boldsymbol{S}_{\omega}x)(t)$ by means of the equality (1.6b), where the integral in the right hand side of (1.6b) is interpreted as a principal value integral.

Under the condition (1.9), the integral $\int\limits_{\mathbb{R}\setminus(t-\varepsilon,t+\varepsilon)}\!\!S_{\omega}(t-\tau)x(\tau)d\tau$ exists as a Lebesgue integral for every $\varepsilon>0$:

[TABLE]

This follows from the estimate (1.10). The assertion that the limit in (1.13) exists for almost every $t\in\mathbb{R}$ will be proved in section 4 using the Hilbert transform theory.

Remark 1.5*.*

Under the assumption (1.9), the function

[TABLE]

which is defined for almost every $t$, is not necessary locally summable. It may happen that $\int\limits_{[a,b]}|y(t)|dt=\infty$ for every finite interval $[a,b]$, $-\infty<a<b<\infty$.

Let us define the transforms $\boldsymbol{\Phi}_{\omega}$ and $\boldsymbol{\Psi}_{\omega}$ formally.

Definition 1.6**.**

For $\boldsymbol{x}(\tau)=\begin{bmatrix}x_{1}(\tau)\\ x_{2}(\tau)\end{bmatrix}\in{}L_{\omega}^{1}\dotplus{}L_{\omega}^{1}$, we put

[TABLE]

where $\boldsymbol{y}(t)$ and $\boldsymbol{z}(t)$ are $2\times 1$ columns:

[TABLE]

with the entries

[TABLE]

The operators $\boldsymbol{C}_{\omega}$ and $\boldsymbol{S}_{\omega}$ are the same that appeared in Lemmas 1.2 and 1.4 respectively.

According to Lemmas 1.2 and 1.4, the values $\boldsymbol{y}(t)$ and $\boldsymbol{z}(t)$ are well defined for almost every $t\in\mathbb{R}$.

The integral transforms (1.1)-(1.3) are said to be the Akhiezer integral transforms.

In another form, these transforms appear in [1, Chapter 15]. (See Problems 3 and 4 to Chapter 15.) The matrix nature of the Akhiezer transforms was camouflaged there.

In what follows, we consider the Akhiezer transform in various functional spaces. We show that the operators $\boldsymbol{\Phi}_{\omega}$ and $\boldsymbol{\Psi}_{\omega}$ are mutually inverse in spaces of functions growing slower than $e^{\omega|t|}$.

2. The operators $\boldsymbol{C}_{\omega}$ and $\boldsymbol{S}_{\omega}$ in $\boldsymbol{L^{2}}$.

The Fourier transform machinery is an adequate tool for study convolution operators.

1. Studing the operators $\boldsymbol{C}_{\omega}$ and $\boldsymbol{S}_{\omega}$ by means of the Fourier transform technique, we deal with the spaces $L^{1}$ and $L^{2}$. Both these spaces consist of measurable functions defined almost everywhere on the real axis $\mathbb{R}$ with respect to the Lebesgue measure. The spaces are equipped by the standard linear operations and the standard norms. If $u\in{}L^{1}$, then

[TABLE]

The space $L^{1}$ consists of all $u$ such that $\|u\|_{L^{1}}<\infty$. If $u\in{}L^{2}$, then

[TABLE]

The space $L^{2}$ consists of all $u$ such that $\|u\|_{L^{2}}<\infty$. This space is equipped by inner product $\langle\,.\,,\,.\,\rangle_{L^{2}}$. If $u^{\prime}\in{}L^{2}$, $u^{\prime\prime}\in{}L^{2}$, then

[TABLE]

2. The Fourier-Plancherel operator $\mathfrak{F}$:

[TABLE]

where

[TABLE]

maps the space $L^{2}$ onto itself isometrically:

[TABLE]

The inverse operator $\mathfrak{F}^{-1}$ is of the form

[TABLE]

where

[TABLE]

Lemma 2.1**.**

Let $f\in{}L^{2}$ and $k\in{}L^{1}$. Then

1. (1)

The integral

[TABLE]

exists as a Lebesgue integral (i.e. $\int\limits_{\mathbb{R}}|k(t-\tau)g(\tau)|d\tau<\infty$) for almost every $t\in\mathbb{R}$. 2. (2)

The function $g$ belongs to $L^{2}$, and the inequality

[TABLE]

holds. 3. (3)

The Fourier-Plancherel transforms $\hat{f}$ and $\hat{g}$ are related by the equality

[TABLE]

where

[TABLE]

This lemma can be found in [2], Theorem 65 there. See also [3], Theorem 3.9.4.

3. Let us calculate the Fourier transforms of the functions $C_{\omega}$ and $S_{\omega}$. The function $C_{\omega}$ belongs to $L^{1}$. So its Fourier transform

[TABLE]

is well defined for every $\lambda\in\mathbb{R}$.

Lemma 2.2**.**

The Fourier transforms $\widehat{C_{\omega}}(\lambda)$ of the function $C_{\omega}(t)$ is:

[TABLE]

The formula (2.14) can be found in [2], where it appears as (7.1.6).

4. The function $S_{\omega}(t)$ does not belong to $L^{1}$. This function has non-integrable singularity at the point $t=0$. Therefore the integral $\int\limits_{\mathbb{R}}S_{\omega}(t)e^{2\pi{}it\lambda}dt$ does not exist as a Lebesgue integral. However $\int\limits_{\mathbb{R\setminus(-\varepsilon,\varepsilon)}}|S_{\omega}(t)|dt<\infty,\,\forall\,\varepsilon>0.$ So the integral

[TABLE]

exists as a Lebesgue integral for every $\varepsilon>0$. We define the Fourier transform $\widehat{S_{\omega}}(\lambda)$ as a principle value integral:

[TABLE]

Lemma 2.3**.**

The limit in (2.16) exists for every $\lambda\in\mathbb{R}$. The Fourier transforms $\widehat{S_{\omega}}(\lambda)$ of the function $S_{\omega}(t)$ is:

[TABLE]

The difference

[TABLE]

satisfies the conditions

[TABLE]

The formula (2.17) can be found in [2], where it appears as (7.2.3).

5. In Section 1 we already have defined the functions $\boldsymbol{C}_{\omega}x$ and $\boldsymbol{S}_{\omega}x$ for $x$ from the space $L^{1}_{\omega}$. The space $L^{2}$ is contained in $L^{1}_{\omega}$. If $x\in{}L^{2}$, then

[TABLE]

According to Lemmas 1.2 and 1.4, if $f\in{}L^{1}_{\omega}$, then the function $(\boldsymbol{C}_{\omega}f)(t)$ is defined for every $t\in\mathbb{R}$ and the function $(\boldsymbol{S}_{\omega}f)(t)$ is defined for almost every $t\in\mathbb{R}$. However for $f\in{}L^{2}(\mathbb{R})$, we can obtain much more accurate results.

Lemma 2.4**.**

Let $f\in{}L^{2}$ and $g=\boldsymbol{C}_{\omega}f$, i.e.

[TABLE]

Then $g\in{}L^{2}$, and the Fourier-Plancherel transforms $\hat{f}$, $\hat{g}$ of functions $f$ and $g$ are related by the equality

[TABLE]

where $\widehat{C_{\omega}}(\lambda)$ is determined by the equality (2.14).

Proof.

Lemma 2.4 is a direct consequence of Lemma 2.1.

∎

Lemma 2.5**.**

Let $f\in{}L^{2}$ and $g=\boldsymbol{S}_{\omega}f$, i.e.

[TABLE]

Then $g\in{}L^{2}$, and the Fourier-Plancherel transforms $\hat{f}$, $\hat{g}$ of functions $f$ and $g$ are related by the equality

[TABLE]

where $\widehat{S_{\omega}}(\lambda)$ is determined by the equality (2.17).

Proof.

Since $S_{\omega}\not\in{}L^{1}$, Lemma 2.5 does not follow from Lemma 2.1 directly. Let

[TABLE]

The function $S_{\omega,\varepsilon}$ belongs to $L^{1}$ for every $\varepsilon>0$. Let

[TABLE]

Applying Lemma 2.1 to $k=S_{\omega,\varepsilon}$, we conclude that $g_{\varepsilon}\in{}L^{2}$ for every $\varepsilon>0$ and that the Fourier-Plancherel transforms $\widehat{g_{\varepsilon}}$, $\widehat{f}$ of the functions $g_{\varepsilon}$, $f$ are related by the equality

[TABLE]

where $\widehat{S_{\omega,\varepsilon}}(\lambda)$ is defined by (2.15). According to Lemma 2.3,

[TABLE]

where

[TABLE]

and the family \{\textup{\LARGE\varrho}_{\omega}(\lambda,\varepsilon)\}_{0<\varepsilon<\infty} satisfies the conditions (2.19) and (2.20). From (2.19), (2.20), (2.30) and the Lebesgue Dominated Convergence Theorem it follows that

[TABLE]

In other words,

[TABLE]

where

[TABLE]

From (2.31) it follows that $\|g_{\varepsilon}-\check{g}\|_{L^{2}}\to 0$ as $\varepsilon\to+0$, i.e.

[TABLE]

where $\check{g}=\mathfrak{F}^{-1}\hat{g}\in{}L^{2}$. From the other side, $(S_{\omega,\varepsilon}f)(t)\to{}g(t)$ for a.e. $t\in\mathbb{R}$ by Lemma 1.4. Hence $g=\check{g}$, and $\hat{g}=\widehat{S_{\omega}}\hat{f}$. ∎

6. The equality

[TABLE]

plays a crucial role in this paper. This equation is a direct consequence of the explicite expressions (2.14) and (2.17) for $\widehat{C_{\omega}}$ and $\widehat{S_{\omega}}$ and the identity

[TABLE]

Lemma 2.6**.**

The operators $\boldsymbol{C}_{\omega}$ and $\boldsymbol{S}_{\omega}$ are contractive in the space $L^{2}(\mathbb{R})$. Moreover the equality

[TABLE]

holds.

Proof.

Let $g_{c}=\boldsymbol{C}_{\omega}f$, $g_{s}=\boldsymbol{S}_{\omega}f$ and let $\hat{f}$, $\widehat{g_{c}}$, $\widehat{g_{s}}$ be the Fourier-Plancherel transforms of the functions $f,\,g_{c},\,g_{s}$. According to Lemmas 2.4 and 2.5, the equalities

[TABLE]

hold. From (2.34) it follows that

[TABLE]

Integrating with respect to $\lambda$, we obtain the equality $\|\widehat{g_{c}}\|^{2}_{L^{2}}+\|\widehat{g_{s}}\|^{2}_{L^{2}}=\|\hat{f}\|^{2}_{L^{2}}$. In view of (2.6), the last equality is equivalent to the equality (2.36). ∎

3. The Akhiezer operators $\boldsymbol{\Phi}_{\omega}$ and $\boldsymbol{\Psi}_{\omega}$ in $\boldsymbol{L^{2}\oplus{}L^{2}}$.

Definition 3.1**.**

The space $L^{2}\oplus{}L^{2}$ is the set of all $2\times 1$ columns $\boldsymbol{x}=\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}$ such that $x_{1}(t)\in{}L^{2}$ and $x_{2}(t)\in{}L^{2}$. The set $L^{2}\oplus{}L^{2}$ is equipped by the natural linear operations and by the inner product $\langle\,,\,\rangle_{L^{2}\oplus{}L^{2}}$.

If $\boldsymbol{x^{\prime}}(t)=\begin{bmatrix}x_{1}^{\prime}(t)\\ x_{2}^{\prime}(t)\end{bmatrix}$ and $\boldsymbol{x^{\prime\prime}}(t)=\begin{bmatrix}x_{1}^{\prime\prime}(t)\\ x_{2}^{\prime\prime}(t)\end{bmatrix}$ belong to $L^{2}\oplus{}L^{2}$, then

[TABLE]

The inner product (3.1) generates the norm

[TABLE]

Since222See (2.21). $L^{2}\subset{}L^{1}_{\omega}$, also $L^{2}\oplus{}L^{2}\subset{}L^{1}_{\omega}\dotplus{}L^{1}_{\omega}$. Thus if $\boldsymbol{x}\in{}L^{2}\oplus{}L^{2}$, then the values $\boldsymbol{y}(t)=(\boldsymbol{\Phi}_{\omega}\boldsymbol{x})(t)$ and $\boldsymbol{z}(t)=(\boldsymbol{\Psi}_{\omega}\boldsymbol{x})(t)$ are defined by (1.16) for almost every $t\in\mathbb{R}$. Using Lemmas 2.4 and 2.5, we conclude from (1.16) that the operators $\boldsymbol{\Phi}_{\omega}$ and $\boldsymbol{\Psi}_{\omega}$ are bounded operators in the space $L^{2}\oplus{}L^{2}$. In particular, the values $\boldsymbol{y}(t)$ and $\boldsymbol{z}(t)$ belong to $L^{2}\oplus{}L^{2}$.

Theorem 3.2**.**

Each of the operators $\boldsymbol{\Phi}_{\omega}$ and $\boldsymbol{\Psi}_{\omega}$ is an isometric operator in the space $L^{2}\oplus{}L^{2}$:

[TABLE]

Theorem 3.3**.**

The operators $\boldsymbol{\Phi}_{\omega}$ and $\boldsymbol{\Psi}_{\omega}$ are mutually inverse in the space $L^{2}\oplus{}L^{2}$:

[TABLE]

Proofs of Theorem 3.2.

Let us associate the $2\times 2$ matrix functions $\widehat{{\varPhi}_{\omega}}(\lambda)$ and $\widehat{{\varPsi}_{\omega}}(\lambda)$ with the operators $\boldsymbol{\Phi}_{\omega}$ and $\boldsymbol{\Psi}_{\omega}$:

[TABLE]

where $\widehat{C_{\omega}}(\lambda)$ and $\widehat{S_{\omega}}(\lambda)$ are the same that in (2.14) and (2.17). Let

[TABLE]

and let $\widehat{\boldsymbol{x}}=\begin{bmatrix}\widehat{x_{1}}\\ \widehat{x_{2}}\end{bmatrix}$, $\widehat{\boldsymbol{y}}=\begin{bmatrix}\widehat{y_{1}}\\ \widehat{y_{2}}\end{bmatrix}$, where $\widehat{x_{1}},\widehat{x_{2}},\widehat{y_{1}},\widehat{y_{2}}$ are the Fourier-Plancherel transforms of the functions $x_{1},x_{2},y_{1},y_{2}$ respectively. According to the equality (1.16) and to Lemmas 2.4 and 2.5, the equality

[TABLE]

holds for almost every $\lambda\in\mathbb{R}$.

From the equality (2.34) it follows that the matrix $\widehat{{\varPhi}_{\omega}}(\lambda)$ is unitary for each $\lambda\in\mathbb{R}$:

[TABLE]

where $I$ is $2\times 2$ identity matrix. From (3.6) and (3.7) it follows that

[TABLE]

i.e.

[TABLE]

Integrating with respect to $\lambda$ over $\mathbb{R}$ and using the Parseval identity (2.6), we conclude that

[TABLE]

that is $\|\boldsymbol{\Phi}_{\omega}\boldsymbol{x}\|_{L^{2}\oplus{}L^{2}}=\|\boldsymbol{x}\|_{L^{2}\oplus{}L^{2}}$. The equality $\|\boldsymbol{\Psi}_{\omega}\boldsymbol{x}\|_{L^{2}\oplus{}L^{2}}=\|\boldsymbol{x}\|_{L^{2}\oplus{}L^{2}}$ can be obtained analogously. ∎

Proof of Theorem 3.3.

Let

[TABLE]

Let $\widehat{\boldsymbol{x}}=\begin{bmatrix}\widehat{x_{1}}\\ \widehat{x_{2}}\end{bmatrix}$, $\widehat{\boldsymbol{y}}=\begin{bmatrix}\widehat{y_{1}}\\ \widehat{y_{2}}\end{bmatrix}$, $\widehat{\boldsymbol{z}}=\begin{bmatrix}\widehat{z_{1}}\\ \widehat{z_{2}}\end{bmatrix}$, where $\widehat{x_{1}},\widehat{x_{2}},\widehat{y_{1}},\widehat{y_{2}},\widehat{z_{1}},\widehat{z_{2}}$ are the Fourier-Plancherel transforms of the functions $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2}$ respectively. We already proved the equality (3.6). In the same way the equality

[TABLE]

can be established. From (3.6) and (3.8) it follows that

[TABLE]

From the equality (2.34) it follows that the matrices $\varPhi_{\omega}(\lambda)$ and $\varPsi_{\omega}(\lambda)$ are mutually inverse:

[TABLE]

where $I$ is $2\times 2$ identity matrix. From (3.9) and (3.10) we conclude that

[TABLE]

Finally $\boldsymbol{z}=\boldsymbol{x}$.

The equality (3.4a) is proved. The equality (3.4b) can be proved in the same way. ∎

4. The Hilbert transform

Definition 4.1**.**

Let $u(\tau)$ be a complex-valued function which is defined for almost every $\tau\in\mathbb{R}$. We assume that the function $u$ satisfies the condition

[TABLE]

Then the integral

[TABLE]

exists for every $t\in\mathbb{R}$ and $\varepsilon>0$. For each $\varepsilon>0$, the function $H_{\varepsilon}u(t)$ is a continuous function of $t$ for $t\in\mathbb{R}$. The function $Hu(t)$ is defined for those $t\in\mathbb{R}$ for which the value $H_{\varepsilon}u(t)$ tends to a finite limit as $\varepsilon\to+0$:

[TABLE]

The function $Hu$ is said to be the Hilbert transform of the function $u$.

Theorem. (A.I.Plessner.) Let $u(\tau)$ be a function which is defined for almost every $\tau\in\mathbb{R}$. If the function $u(\tau)$ satisfies the condition (4.1), then its Hilbert transform $Hu(t)$ exists for almost every $t\in\mathbb{R}$.

Proof of this Plessner’s Theorem can be found in [2], Theorem 100 there.

If $u$ is a function from $L^{2}$, then $u$ satisfies the condition (4.1). By Plessner’s theorem, the Hilbert transform $v(t)=(Hu)(t)$ exists for almost every $t\in\mathbb{R}$.

Theorem. (E.C. Titchmarch.) Let $u$ be a function from $L^{2}$. Then:

1. (1)

Its Hilbert transform $v=Hu$ also belongs to $L^{2}$, and the equality

[TABLE]

holds. 2. (2)

The equality

[TABLE]

holds for almost every $t\in\mathbb{R}$.

This theorem means that the Hilbert transform, considered as an operator in $L^{2}$, is an unitary operator which satisfies the equality

[TABLE]

where ${I}$ is the identity operator in $L^{2}$.

Proof of Lemma 1.4..

We use the decomposition (1.11) of the kernel $S(t-\tau)$ into the sum of the Hilbert kernel $\dfrac{1}{\pi(t-\tau)}$ and the ‘regular’ kernel $r(t-\tau)$. Let $(a,b)\subset\mathbb{R}$ be an arbitrary finite interval of the real axis. We split the function $f(\tau)$ into the sum of two summands.

[TABLE]

So

[TABLE]

According to (1.11) and (4.7), the equality

[TABLE]

holds, where

[TABLE]

The function $g$ satisfies the condition (4.1). According Plessner’s Theorem, $\lim\limits_{\varepsilon\to+0}I_{1,\varepsilon}(t)$ exists for almost every $t\in\mathbb{R}$. Since the function $g$ is finitely supported and the kernel $r(t-\tau)$ is continuous, $\lim\limits_{\varepsilon\to+0}I_{2,\varepsilon}(t)$ exists for every $t\in\mathbb{R}$. Since the function $h(\tau)$ vanishes for $\tau\in(a,b)$ and $\int\limits_{\mathbb{R}}\frac{|h(\tau)|}{\cosh{}\omega\tau}d\tau<\infty$, $\lim\limits_{\varepsilon\to+0}I_{3,\varepsilon}(t)$ exists for every $t\in(a,b)$. In view of (4.10), the limit in (1.13) exists for almost every $t\in(a,b)$. Since $(a,b)$ is an arbitrary finite interval, the limit in (1.13) exists for almost every $t\in\mathbb{R}$. ∎

5. The operators $\boldsymbol{C}_{\omega}$ and $\boldsymbol{S}_{\omega}$ in $\boldsymbol{L^{2}_{\sigma}}$.

In this section we consider the operators $\boldsymbol{C}_{\omega}$ and $\boldsymbol{S}_{\omega}$ acting in spaces of functions growing slower than $e^{\omega|t|}$ as $t\to\pm\infty$.

Definition 5.1**.**

For $\sigma\in\mathbb{R}$, the space $L^{2}_{\sigma}$ is the space of all functions $x$ which are measurable, defined almost everywhere with respect to the Lebesgue measure and satisfy the condition $\|x\|_{L^{2}_{\sigma}}<\infty$, where

[TABLE]

The space $L^{2}_{\sigma}$ is equipped by the standard linear operations and by the norm (5.1).

It is clear that the space $L^{2}$ which appeared in section 2 is the space $L^{2}_{{}_{0}}$, that is $L^{2}_{\sigma}$ with $\sigma=0$.

In Section 1 we already have defined the functions $\boldsymbol{C}_{\omega}x$ and $\boldsymbol{S}_{\omega}x$ for $x$ from the space $L^{1}_{\omega}$. For $\sigma<\omega$, the space $L^{2}_{\sigma}$ is contained in $L^{1}_{\omega}$. If $x\in{}L^{2}_{\sigma}$, then

[TABLE]

According to Lemmas 1.2 and 1.4, if $f\in{}L^{1}_{\omega}$, then the function $(\boldsymbol{C}_{\omega}f)(t)$ is defined for every $t\in\mathbb{R}$ and the function $(\boldsymbol{S}_{\omega}f)(t)$ is defined for almost every $t\in\mathbb{R}$.

In Section 2 we obtained that if $f\in{}L^{2}$, than $\boldsymbol{C}_{\omega}f\in L^{2}$ and $\boldsymbol{S}_{\omega}f\in L^{2}$. Moreover we proved that the operators $\boldsymbol{C}_{\omega}$ and $\boldsymbol{S}_{\omega}$ are contractive in $L^{2}$: see Corollary 2.6. In this section we show that if $0<\sigma<\omega$ and $f\in{}L^{2}_{\sigma}$, than $\boldsymbol{C}_{\omega}f\in L^{2}_{\sigma}$ and $\boldsymbol{S}_{\omega}f\in L^{2}_{\sigma}$. Moreover we show that the operators $\boldsymbol{C}_{\omega}$ and $\boldsymbol{S}_{\omega}$ are bounded in the space $L^{2}_{\sigma}$.

Lemma 5.2**.**

Assume that $0\leq\sigma<\omega$. Let $f\in{}L^{2}_{\sigma}$, and $g$ is related to $f$ by means of the formula (2.22), i.e. $g=\boldsymbol{C}_{\omega}f$. Then $g\in{}L^{2}_{\sigma}$, and

[TABLE]

where $M_{c}<\infty$ is a value which does not depend on $\omega$ and $\sigma$.

Proof.

Let

[TABLE]

Since $f\in{}L^{2}_{\sigma}$, $u\in{}L^{2}$. The equality (2.22) can be rewritten as

[TABLE]

Let us estimate the kernel

[TABLE]

For $\sigma\geq 0$ the inequality

[TABLE]

holds. Hence

[TABLE]

From this inequality and from the expression (1.4) for $C_{\omega}$ we conclude that

[TABLE]

where

[TABLE]

For $0\leq\sigma<\omega$, the function $k^{c}_{\sigma,\omega}$ belongs to $L^{1}$ and

[TABLE]

where

[TABLE]

The integral in (5.11) can be calculated explicitly:

[TABLE]

Thus

[TABLE]

Since $\sin\frac{\pi}{2}\eta\geq\eta$ for $0\leq\eta\leq 1$, the inequality (5.14) implies the inequality

[TABLE]

From (5.5) and (5.9) we obtain the inequality $|v(t)|\leq w(t),\ \forall\,t\in\mathbb{R}$, and

[TABLE]

where

[TABLE]

According to Lemma 2.1, $w\in{}L^{2}$, and the inequality

[TABLE]

holds. The inequality

[TABLE]

is a consequence of the equalities (5.16), (5.18) and (5.15). According to (5.4), $\|u\|_{L^{2}}=\|f\|_{L^{2}_{\sigma}}$, $\|v\|_{L^{2}}=\|g\|_{L^{2}_{\sigma}}$. So the inequality (5.3) holds with $M_{c}=2$. ∎

Lemma 5.3**.**

Assume that $0\leq\sigma<\omega$. Let $f\in{}L^{2}_{\sigma}$, and $g$ is related to $f$ by means of the formula (2.24), i.e. $g=\boldsymbol{S}_{\omega}f$. Then $g\in{}L^{2}_{\sigma}$, and

[TABLE]

where $M_{s}<\infty$ is a value which does not depend on $\omega$ and $\sigma$.

Proof.

Let $u(\tau)$, $v(t)$ be defined according to (5.4). Since $f\in{}L^{2}_{\sigma}$, $u\in{}L^{2}$. The equality (2.24) can be rewritten as

[TABLE]

We present $v(t)$ as

[TABLE]

where

[TABLE]

[TABLE]

Let us estimate the kernel

[TABLE]

From the inequalities $|e^{\xi}-1|\leq|\xi|e^{|\xi|}$, from (5.7) and from the expression (1.4) for $S_{\omega}$ we conclude that

[TABLE]

where

[TABLE]

The function $k^{s}_{\sigma,\omega}$ belongs to $L^{1}$, and

[TABLE]

where $a$ is the same that in (5.12). The integral in (5.27) can be calculated explicitly:

[TABLE]

Thus the inequality

[TABLE]

holds. From (5.23), (5.24) and (5.25) it follows that

[TABLE]

where

[TABLE]

According to Lemma 2.1, $w\in{}L^{2}$, and the inequality

[TABLE]

holds. From (5.29), (5.32) and (5.30) we conclude that

[TABLE]

According to Lemma 2.6, the inequality

[TABLE]

holds. From (5.21), (5.34) and (5.33) we derive the inequality

[TABLE]

with $M_{s}=\pi+1$. ∎

6. The Akhiezer operators $\boldsymbol{\Phi}_{\omega}$ and $\boldsymbol{\Psi}_{\omega}$ in $\boldsymbol{L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}}$.

Definition 6.1**.**

The space $L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$ is the set of all $2\times 1$ columns $\boldsymbol{x}=\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}$ such that $x_{1}(t)\in{}L^{2}_{\sigma}$ and $x_{2}(t)\in{}L^{2}_{\sigma}$, where $L^{2}_{\sigma}$ was defined in Definition 5.1. The set $L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$ is equipped by the natural linear operations and by the norm

[TABLE]

Since333See (5.2). $L^{2}_{\sigma}\subset{}L^{1}_{\omega}$, also $L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}\subset{}L^{1}_{\omega}\dotplus{}L^{1}_{\omega}$. Thus if $\boldsymbol{x}\in{}L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$, then the values $\boldsymbol{y}(t)=(\boldsymbol{\Phi}_{\omega}\boldsymbol{x})(t)$ and $\boldsymbol{z}(t)=(\boldsymbol{\Psi}_{\omega}\boldsymbol{x})(t)$ are defined by (1.16) for almost every $t\in\mathbb{R}$. From Lemmas 5.2 and 5.3 we derive

Lemma 6.2**.**

We assume that $0\leq\sigma<\omega$. Let $\boldsymbol{x}\in{}L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$ and let $\boldsymbol{y}=\boldsymbol{\Phi}_{\omega}\boldsymbol{x},$ $\boldsymbol{z}=\boldsymbol{\Psi}_{\omega}\boldsymbol{x}$ be defined by (1.16). Then $\boldsymbol{y}\in{}L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$, $\boldsymbol{z}\in{}L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$, and the estimates hold

[TABLE]

where $M<\infty$ is a value which does not depend on $\sigma,\,\omega,\,\boldsymbol{x}$.

The following theorem is a main result of this paper.

Theorem 6.3**.**

We assume that $0\leq\sigma<\omega$. Then for every $\boldsymbol{x}\in{}L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$ the equalities

[TABLE]

hold.

Proof.

From Lemma 6.2 it follows that the operators $\boldsymbol{\Psi}_{\omega}\boldsymbol{\Phi}_{\omega}$ and $\boldsymbol{\Phi}_{\omega}\boldsymbol{\Psi}_{\omega}$ are bounded linear operators in the space $L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$. The set $L^{2}\oplus{}L^{2}$ is a dense subset of the space $L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$. By Theorem 3.3, the equalities (6.4) holds for every $\boldsymbol{x}\in{}L^{2}\oplus{}L^{2}$. By continuity, the equalities (6.4) can be extended from $L^{2}\oplus{}L^{2}$ to $L^{2}_{\sigma}\oplus{}L^{2}_{\sigma}$. ∎