1 Introduction

Convolution operations are very important in mathematical community, as well as in signal processing, sampling, filter design and applications. Lax et al. (see [1]–[3]) considered the equation

$$\begin{aligned} \int \limits _ {-\infty }^0 f (u + \tau ) g (u) du=0,\, \tau \le 0,\quad g \in L^2 (-\infty ; 0), \end{aligned}$$
(1.1)

where unknown function f belongs to \(L^2 (-\infty ; 0)\). For the analysis of this equation the following characteristic functions

$$\begin{aligned} G (z)= & {} \frac{1}{i \sqrt{2 \pi }} \int \limits _ {-\infty }^{0} g (u) e^{uz} du, \end{aligned}$$
(1.2)
$$\begin{aligned} F (z)= & {} \frac{1}{\sqrt{2 \pi }} \int \limits _ {-\infty }^{0} f (u) e^{-zu} du \end{aligned}$$
(1.3)

was used. Equation (1.1) has a number of direct applications in theory of translation invariant subspaces, cyclicity problem, spectral analysis-synthesis problem; also this equation is of interest as a model case for the study of locally compact abelian groups (see [3, 19]). In [32] the author obtained the criterion of the Bedrosian identity in terms of equation (1.1). Therefore generalizations of equation (1.1) have a good perspective in applications.

Let \(H^p (\mathbb C _ +), 1 \le p <+ \infty , \) be the Hardy space of analytic in the half-plane \(\mathbb C _ +: = \{z: \mathfrak {R}z> 0 \} \) functions for which

$$\begin{aligned} || f ||_* = \sup \limits _ {x> 0} \left\{ \int \limits _ {-\infty }^{+ \infty } |f (x+iy) |^p dy \right\} ^{1/p} <+ \infty . \end{aligned}$$

Properties of these spaces are described in details in [18]. Each function \(\psi \in H^p (\mathbb C _ +), 1 \le p <+ \infty , \) has nontangential boundary values almost everywhere on the imaginary axis \(i\mathbb R=\partial \mathbb C _ +\). These values we also define by \(\psi \) and \(\psi \in L^p(i\mathbb R)\).

N. Wiener and R. Paley [23] proved that (1.2) is a bijection of \(H^2 (\mathbb C _ +)\ni G \) onto \(L^2(-\infty ;0)\ni g \). It is clear, equality (1.3) is an bijection of \(L^2(-\infty ;0)\ni f \) onto \(H^2 (\mathbb C _ -)\ni F \), where \(H^2 (\mathbb C _ -) \) is the Hardy space on \(\mathbb C _ -: = \{z: \mathfrak {R}z< 0 \} \).

Let us define the function \(\Phi (iy)=F(iy)G(iy)\) almost everywhere on \(i\mathbb R\), where F(iy) and G(iy) are nontangential boundary values of the functions F and G. The following solvability criterion (see, for example [3], p. 89) is the fundamental fact for the analysis of Eq. (1.1).

Theorem 1.1

[1] \(f\in L^2 (-\infty ; 0)\) is a solution of Eq. (1.1) if and only if there exists a function \(P\in H^1 (\mathbb C _ +)\) such that nontangential boundary values of P equal to \(\Phi \) almost everywhere on \(i\mathbb R\).

While often applied to real-valued signals, convolution can be used on complex signals. Many types of signals, such as seismic data or electrical signals, contain significant information in the phase of the signal [10, 11].

Some analogues of Theorem 1.1 for Hardy-type spaces on angular domain is proved in [9]. But for the case of a half-strip domain another methods is required: characteristic functions of functions in the Hardy (Smirnov) spaces on an angular domain belong to Hardy spaces on (generally other) angular domain; characteristic functions of functions in the Hardy (Smirnov) spaces on an half-strip belong to weighted Hardy spaces on half-plane.

The aim of this paper is to obtain an equivalent result to Theorem 1.1 for the convolution equation on half-strip. This equation has direct applications to signal theory [24], to the studies of cyclic functions [26]. But for the studies of translation invariant subspaces [19] and some integral operators the full analogue of Theorem 1.1 is needed. In [25] only some partial results was obtained. The case of half-strip is interest as a model case for the study of equations of convolution type, for study of nonselfadjoint operators, Toeplitz operators, locally compact groups (see [4]–[8, 13, 27, 28]).

2 The Main Result

First we need some definitions of analogues of the space \(L^2(-\infty ;0)\). Let\(D _ {\sigma } = \{z: \mathfrak {R}z<0,|\mathfrak {I}z| <\sigma \} \), \(D _ {\sigma }^* = \mathbb C \backslash {\overline{D}} _ {\sigma }, \, \sigma >0 \). Let \(E^p [D _ {\sigma }], 1 \le p <+ \infty , \) be the space of analytic functions on \(D _ {\sigma } \) for which

$$\begin{aligned} \sup \left\{ \int \limits _ \gamma |f (z) |^p |dz | \right\} ^{1/p} <+ \infty , \end{aligned}$$
(2.1)

where the supremum is over all segments \( \gamma \) which lay in \(D _ {\sigma } \) and let \(E^p_* [D _{\sigma }] \) be the space of analytic functions f on \(D_ {\sigma }^* \) for which condition (2.1) holds, where the supremum is over all segments \( \gamma \) which lay in \(D _ {\sigma }^* \). Functions f belonging to these spaces have almost everywhere on \( \partial D_ \sigma \) nontangential boundary values, which we denote by f(z) and \(f \in L^p [\partial D_ \sigma ] \). In fact, \(E^p [D _{\sigma }]\) and \(E^p_* [D _{\sigma }]\) are the Hardy spaces in their domains.

Sedletskii [17] proved that the space \(H^p (\mathbb C _ +)\) can be defined as the class of analytic on \(\mathbb C _ +\) functions for which

$$\begin{aligned} || f ||: = \sup \limits _ {-\frac{\pi }{2}<\varphi<\frac{\pi }{2}} \left\{ \int \limits _0^{+ \infty } | f (re^{i \varphi }) |^p dr \right\} ^{1/p} <+ \infty . \end{aligned}$$

Therefore Vynnytskyi [20] has considered the following generalization of the Hardy space. Let \(H^p_ \sigma (\mathbb C _ +), \sigma \ge 0,1 \le p <+ \infty , \) be the space of analytic in \( \mathbb C _ + \) functions for which

$$\begin{aligned} || f ||: = \sup \limits _ {-\frac{\pi }{2}<\varphi<\frac{\pi }{2}} \left\{ \int \limits _0^{+ \infty } | f (re^{i \varphi }) |^p e^{-pr \sigma | \sin \varphi |} dr \right\} ^{1/p} <+ \infty . \end{aligned}$$

In [21] the following generalization of equation (1.1)

$$\begin{aligned} \int \limits _ {\partial D_ \sigma } f (w + \tau ) g (w) dw=0, \,\tau \le 0,\quad g \in E^2_* [D_ \sigma ], \end{aligned}$$
(2.2)

is considered, where an unknown function f belongs to \(E^2 [D_ \sigma ]\). The inclusion \(f\in E^2 [D_\sigma ],\) \(\sigma >0,\) implies that f is defined almost everywhere on \(\overline{D}_\sigma =\{z: \mathfrak {R}z \le 0,|\mathfrak {I}z| \le \sigma \}\) and \(f\in L^2( \partial D_\sigma )\). Therefore it is natural to define \(D_0:=(-\infty ;0)\), \(E^2 [D_0]=L^2(-\infty ;0)\) and \(E^2_* [D_0]=L^2(-\infty ;0)\) as well. Hence for the limit case \(\sigma =0\) the above equation is equal to (1.1).

The set of solutions of Eq. (1.1) is significantly different [26] from (1.2). By [21], each of the equalities

$$\begin{aligned} G (z) = \frac{1}{i \sqrt{2 \pi }} \int \limits _ {\partial D_ \sigma } g (w) e^{zw} dw \end{aligned}$$
(2.3)

and

$$\begin{aligned} g (w) = \frac{1}{\sqrt{2 \pi }} \int _0^{+ \infty } G (x) e^{-xw} dx, \qquad \mathfrak {R}\, w> 0. \end{aligned}$$
(2.4)

is the bijection of \(H^2_ \sigma (\mathbb C _ +) \) onto \(E^2_ * [D_\sigma ] \).

Let \(\sigma > 0\), \(p>0\). We define the Paley - Wiener space \(W_\sigma ^p\) as the space of entire functions, whose restriction on \(\mathbb R\) belongs to \(L^p(\mathbb R)\) and which satisfy the inequality \(|f(z)|<c e^{\sigma |z|}\) for a positive constant C.

Let \(T^2_ \sigma (\mathbb C_-) \) be a set of points \(F = (F_1, F_2, F_3) \), where \(F_1 (z) e^{-i \sigma z} \in H^2 (\mathbb C_-) \), \(F_3 (z) e^{i \sigma z} \in H^2 (\mathbb C_-), \) \(F_2 \in W^2_\sigma \) , and \(F_1 (z) +F_2 (z) +F_3 (z) \equiv 0 \) for \(z \in \mathbb C_- = \{\mathfrak {R}z <0 \} \).

Suppose \(l_1=\{w=u-i\sigma : -\infty <u\le 0\},\) \(l_2= [-i \sigma ; i \sigma ] \), and \(l_3=\{w=u+i\sigma : -\infty <u\le 0\} \); \(l_j\) are oriented anticlockwise with respect to \(D_\sigma \).

Lemma 2.1

[21] Equalities

$$\begin{aligned} F_j (z) = \frac{1}{\sqrt{2 \pi }} \int _ {l_j} f (w) e^{-zw} dw, \qquad f \in E^2 [D_ \sigma ], \qquad j \in \{1,2,3 \}, \end{aligned}$$
(2.5)

are the bijection of \(T^2_ \sigma (\mathbb C_-) \) onto \(E^2 [D_ \sigma ]\).

Each function \(f\in W_{\sigma }^2\) can be represented as cardinal series by Whittaker–Kotelnikov–Shannon–Nyquist formula:

Lemma 2.2

([12, 14, 15]) The space \(W_\sigma ^2\) coincides with the space of functions represented as

$$\begin{aligned} f(t)=\sigma \sqrt{\frac{2}{\pi }}\sum \limits _{k=-\infty }^{+\infty }c_k\frac{\sin \sigma t}{\sigma t-\pi k}, \qquad (c_k)\in l^2. \end{aligned}$$
(2.6)

The space \(W_\sigma ^1\) coincides with the space of functions represented as (2.6) such that \((c_k)\in l^1,\)

$$\begin{aligned} \sum \limits _{m=-\infty }^{+\infty }\left| \sum \limits _{k=-\infty }^{+\infty } (-1)^{k+m}c_{k+m}\frac{k}{1+k^2}\right| <+\infty . \end{aligned}$$
(2.7)

The space \(W_\sigma ^p\) can be defined also [16] as the space of entire functions satisfying the condition

$$\begin{aligned} \sup \limits _{\varphi \in (0;2\pi )}\left\{ \int \limits _{0}^{+\infty }|(fre^{i\varphi })|^pe^{-p\sigma r|\sin \varphi |}dr\right\} ^{1/p}<+\infty . \end{aligned}$$

It is clear that \(H^p_ \sigma (\mathbb C _ +)\) is a weighted Hardy space and is an analogue of the Paley-Wiener space for the half-plane as well.

Let \(\Phi _j (it)=F_j (it)G(it),\) \(j\in \overline{1;3},\) \(t\in {\mathbb R}\). The main result of this paper is the following.

Theorem 2.1

A function \(f\in E^2 [D_\sigma ] \) is a solution of equation (2.2) if and only if there exists a function \(P_1\), \(P_1e^{-i\sigma z}\in H^1_\sigma (\mathbb C _ +)\), such that nontangential boundary values of \(P_1\) equal to \(\Phi _1(it)\), \(t\in \mathbb R,\) almost everywhere on the imaginary axis.

Clearly, Theorem 1.1 is a partial case of Theorem 2.1.

Otherwise, Theorem 2.1 can be reformulate as

Theorem 2.2

A function \(f\in E^2 [D_\sigma ] \) is a solution of equation (2.2) if and only if there exists a function \(P_3\), \(P_3e^{-i\sigma z}\in H^1_\sigma (\mathbb C _ +)\), such that nontangential boundary values of \(P_3\) equal to \(\Phi _3(it)\), \(t\in \mathbb R,\) almost everywhere on the imaginary axis.

3 Proof of the Main Result

Sufficiency is proved in [25]. The proof of necessity is based on the following convolution theorem.

Lemma 3.1

[21] If \(g\in E_*^2 [D_\sigma ]\), \(f\in E^2[D_\sigma ],\) then for all \(\tau \le 0\) the following equality

$$\begin{aligned} \int \limits _{\partial D_\sigma } {f(w+\tau )g(w)dw=\int \limits _0^{+i\infty } {\Phi _1 (z)} e^{\tau z}dz+} \int \limits _{-i\infty }^0 {\Phi _3 (z)} e^{\tau z}dz+\int \limits _0^{+\infty } {\Phi _2 (z)} e^{\tau z}dz,\nonumber \\ \end{aligned}$$
(3.1)

is valid.

The main idea of the following proof is the reduction of properties of the weighted Hardy space to the classical Hardy space.

Proof of Theorem 2.1

Note that \(F_2 \in W^2_\sigma \) by the Paley-Wiener Theorem. Hence \(\Phi _2 \in H^1_{2\sigma } (\mathbb C_+)\) by Hölder’s inequality. Suppose

$$\begin{aligned}&S(z)=-\frac{1}{2\pi i}\int \limits _0^{+i\infty } {\Phi _1 (w)\frac{1}{w-z}dw} +\frac{1}{2\pi i}\int \limits _{-i\infty }^0 {\Phi _3 (w)\frac{1}{w-z}} dw\\&-\frac{1}{2\pi i}\int \limits _0^{+\infty } {\Phi _2 (w)\frac{1}{w-z}dw}. \end{aligned}$$

Let \(E^p[\mathbb C(\alpha ;\gamma )], 0<\gamma -\alpha<2\pi , 1\le p<+\infty \), be the space of analytic functions f in \(\mathbb C(\alpha ;\gamma )=\{z:\gamma<\arg z<\beta \}\) for which

$$\begin{aligned} \sup _{\alpha<\varphi<\gamma }\left\{ \int _{0}^{+\infty }|f(re^{i\varphi })|dr\right\} <+\infty . \end{aligned}$$

Functions \(f\in E^p[\mathbb (\alpha ;\gamma )]\) have almost everywhere on \(\partial \mathbb C(\alpha ;\gamma )\) [9] the angular boundary values, which we denote by f and \(f\in L^p[\mathbb C(\alpha ;\gamma )]\).

The first step. We claim the function

$$\begin{aligned} P_1(z)=\left\{ {{\begin{array}{*{20}l} {S(z),z\in {\mathbb C}(0;\pi /2),} \\ {S(z)-\Phi _2 (z),z\in {\mathbb C}(-\pi /2;0)} \\ \end{array} }} \right. \end{aligned}$$

is the analytic continuation of \(\Phi _1\) from \(i\mathbb R\) to \(\mathbb C_+.\) We designate by \(\eta (\tau )\) the right hand side of (3.1). Since

$$\begin{aligned} \int \limits _{-\infty }^0 \eta (\tau ) e^{\tau \zeta } d\tau =0,\qquad \mathfrak {R}\zeta <0, \end{aligned}$$

by Fubini theorem we have

$$\begin{aligned} S(z)=0 \text { for all } z\in \mathbb C_-. \end{aligned}$$
(3.2)

Hence for \(z\in \mathbb C_+\)

$$\begin{aligned} S(z)=S(z)-S(-\overline{z})= \frac{1}{\pi }\int \limits _0^{\infty }\frac{x\Phi _1 (iv)}{(v-y)^2+x^2}dv-\frac{1}{\pi }\int \limits _{-\infty }^0 \frac{x\Phi _3 (iv)}{(v-y)^2+x^2}dv\\ -\frac{1}{\pi i}\int \limits _0^{+\infty }\frac{\Phi _2 (u)x}{(u-x-iy)(u+x-iy)}du. \end{aligned}$$

The first integral in the left hand side of the above equality is the Poisson integral for the function

$$\begin{aligned} \psi _1(y):=\left\{ {{\begin{array}{*{20}l} {\Phi _1 (iy),\,y>0,} \\ {0,\,y<0}, \\ \end{array} }} \right. \end{aligned}$$

therefore the nontangential boundary values from \(\mathbb C_+\) of this integral are equal to \(\psi _1(y)\) almost everywhere on \(i\mathbb R.\) Analogously the nontangential boundary values from \(\mathbb C_+\) of the second integral are equal to

$$\begin{aligned} \psi _3(y):=\left\{ {{\begin{array}{*{20}l} {\Phi _3 (iy),\,y<0,} \\ {0,\,y>0}, \\ \end{array} }} \right. \end{aligned}$$

almost everywhere on \(i\mathbb R.\) The third integral tends to zero as x tends to zero. Hence the nontangential boundary values of S(z) are equal to

$$\begin{aligned} \psi (y):=\left\{ {{\begin{array}{*{20}c} {\Phi _1 (iy),\,y>0,} \\ {-\Phi _3 (iy),\,y<0}. \\ \end{array} }} \right. \end{aligned}$$

Since \(\Phi _1\equiv -\Phi _2 -\Phi _3 \), we have that nontangential boundary values of \(P_1\) equal to \(\Phi _1(it)\), \(t\in \mathbb R,\) almost everywhere on \(i\mathbb R\).

It remains to prove \(P_1(z)e^{-i\sigma z}\in H^1_\sigma (\mathbb C _ +)\).

The second step. There exist analytic in \(\mathbb C_+\) functions \(\beta _1\), \(\beta _2\) such that \(\Phi _2=\beta _1-\beta _2\), \(\beta _1\in E^1[\mathbb C(0;\pi /2)]\), \(\beta _2\in E^1[\mathbb C(-\pi /2;0)]\).

It is easy to see

$$\begin{aligned} \Phi _2(z)=F_2(z)G(z)=-\frac{1}{\sqrt{2 \pi }}F_2(z)\left( \int _ {l_1} g (w) e^{zw} dw+\int _ {l_3} g (w) e^{zw} dw\right. \\ \left. +i\int _{-\sigma }^\sigma g (iv) e^{izv} dv\right) = -\frac{1}{\sqrt{2 \pi }}F_2(z)\left( e^{-i\sigma z}\int \limits _{-\infty }^0 g(u-i\sigma )e^{uz}du \right. \\ \quad \left. -e^{i\sigma z}\int \limits _{-\infty }^0 g(u+i\sigma )e^{uz}du +i\int _{-\sigma }^\sigma g (iv) e^{izv} dv \right) \\ =F_2(z)\left( e^{-i\sigma z} h_1(z)+ e^{i\sigma z}h_3(z)+h_2(z)\right) . \end{aligned}$$

By Paley-Wiener Theorem [18] \(h_1\in H^2(\mathbb C_+)\), \(h_3\in H^2(\mathbb C_+)\). Therefore \(F_2(z) h_3(z)e^{i\sigma z}\in E^1[\mathbb C(0;\pi /2)]\), \(F_2(z) h_1(z)e^{-i\sigma z}\in E^1[\mathbb C(-\pi /2;0)]\). By another Paley-Wiener Theorem [18] \(h_2\in W^2_\sigma .\) It remains to show that \(\Phi _{22}:=F_2 h_2\) has an appropriate decomposition as \(\Phi _2\).

The third step. There exist analytic in \(\mathbb C_+\) functions \(\mu _1\), \(\mu _2\) such that \(\Phi _{22}=\mu _1-\mu _2\), \(\mu _1\in E^1[\mathbb C(0;\pi /2)]\), \(\mu _2\in E^1[\mathbb C(-\pi /2;0)]\).

Using Hölder’s inequality, it is easy to see that \(\Phi _{22}\in W^1_{2\sigma }.\) By Lemma 2.2 we have

$$\begin{aligned} \Phi _{22}(t)=\sum \limits _{k=-\infty }^{+\infty }b_k\frac{\sin 2\sigma t}{2\sigma t-\pi k}, \qquad (b_k)\in l^1 \end{aligned}$$

and

$$\begin{aligned} \sum \limits _{m=-\infty }^{+\infty }\left| \sum \limits _{k=-\infty }^{+\infty } (-1)^{k+m}b_{k+m}\frac{k}{1+k^2}\right| <+\infty . \end{aligned}$$

Therefore

$$\begin{aligned} \Phi _{22}(z)=\sum \limits _{n=-\infty }^{+\infty }b_{2n}\frac{\sin 2\sigma z}{2\sigma z-2\pi n}+b_{2n+1}\frac{\sin 2\sigma z}{2\sigma z-2\pi n-\pi }=\psi (z)+\eta (z), \end{aligned}$$

where

$$\begin{aligned} \psi (z) =\frac{1}{\pi }\sum \limits _{n=-\infty }^{+\infty }\frac{-b_{2n}\sin 2\sigma z}{(2\sigma z-2\pi n)(2\sigma z-2\pi n-\pi )}, \end{aligned}$$
(3.3)
$$\begin{aligned} \eta (z) =\frac{1}{\pi }\sum \limits _{n=-\infty }^{+\infty }\frac{(b_{2n}+b_{2n+1})\sin 2\sigma z}{2\sigma z-2\pi n-\pi }. \end{aligned}$$
(3.4)

First of all we prove that \(\psi \in W_{2\sigma }^1\). Since series in (3.3) converges absolutely and uniformly on each compact set in \(\mathbb C_+\), we have

$$\begin{aligned}&\int \limits _{-\infty }^{+\infty }|\psi (x+i)|dx\le d_1\sum \limits _{k=-\infty }^{+\infty }|b_{2k}|\left( \int \limits _{-\infty }^{-2\pi k-2\pi }\frac{ dx}{|2\sigma x+2\sigma i-2\pi k||2\sigma x+2\sigma i-2\pi k-\pi |}\right. \\&+\int \limits _{-2\pi k-2\pi }^{-2\pi k+2\pi }\frac{ dx}{|2\sigma x+2\sigma i-2\pi k||2\sigma x+2\sigma i-2\pi k-\pi |}\\&+\left. \int \limits _{-2\pi k+\pi }^{+\infty }\frac{ dx}{|2\sigma x+2\sigma i-2\pi k||2\sigma x+2\sigma i-2\pi k-\pi |}\right) \\\le & {} d_1 \sum \limits _{k=-\infty }^{+\infty }|b_{2k}|\left( \int \limits _{-\infty }^{-2\pi k-2\pi }\frac{1}{\sqrt{(2\sigma x-2\pi k)^2+4\sigma ^2}\sqrt{(2\sigma x-2\pi k)^2+4\sigma ^2}}dx\right. \\&+\left. \int \limits _{-2\pi k-2\pi }^{-2\pi k+2\pi }\frac{1}{4\sigma } dx+\int \limits _{-2\pi k+2\pi }^{+\infty }\frac{1}{\sqrt{(2\sigma x-2\pi k-\pi )^2+4\sigma ^2}\sqrt{(2\sigma x-2\pi k-\pi )^2+4\sigma ^2}}dx\right) \\\le & {} d_2\sum \limits _{k=-\infty }^{+\infty }|b_{2k}|<+\infty , \end{aligned}$$

where \(d_1>0\), \(d_2>0.\)

Since \(\psi (x+i)\in L^1 (\mathbb R)\), it follows from definition that \(\psi (z+i)\in W_{2\sigma }^{1}\). Therefore \(\psi (z)\in W_{2\sigma }^{1}\) and \(\eta =f-\psi \in W_{2\sigma }^1\). It follows from (3.4) that

$$\begin{aligned} \eta (z)=\frac{1}{\pi }\sum \limits _{n=-\infty }^{+\infty }\frac{(b_{2n}+b_{2n+1})\sin 2\sigma z}{2\sigma z-2\pi n-\pi }=\mu _{11}-\mu _{21}, \end{aligned}$$

where

$$\begin{aligned} \mu _{11}(z)=\sum \limits _{k=-\infty }^{+\infty }\frac{(e^{2i\sigma z}+1)(b_{2n}+b_{2n+1})}{2\pi i(2\sigma z-2\pi n-\pi )},\\ \mu _{21}(z)=\sum \limits _{k=-\infty }^{+\infty }\frac{(e^{-2i\sigma z}+1)(b_{2n}+b_{2n+1})}{2\pi i(2\sigma z-2\pi n-\pi )}. \end{aligned}$$

Each series above converges absolutely and uniformly on each compact set in \(\mathbb C\) because sequence \((b_{2n}+b_{2n+1} )\) belongs to \(l_1\). Therefore both the functions \(\mu _{11}\) and \(\mu _{21}\) are entire functions of exponential type \(\le \sigma \). Let us remark that

$$\begin{aligned} \mu _{11}(z)&=\sum \limits _{k=-\infty }^{+\infty }\frac{e^{{i\sigma z}}(e^{i\sigma z}+e^{-i\sigma z})(b_{2n}+b_{2n+1} )}{2\pi i(2\sigma z-2\pi n-\pi )}\\&=\sum \limits _{k=-\infty }^{+\infty }\frac{(b_{2n}+b_{2n+1})\sin 2\sigma z}{\pi (2\sigma z-2\pi n-\pi )}\times \frac{e^{{i\sigma z}}(e^{i\sigma z}+e^{-i\sigma z})}{2i\sin 2\sigma z}=\eta (z)\frac{ e^{{i\sigma z}}}{ i\sin {\sigma z}}. \end{aligned}$$

Since on the line \(z=x+i\) the function

$$\begin{aligned} \frac{ e^{{i\sigma z}}}{ i\sin {\sigma z}} \end{aligned}$$

is bounded and \(\eta \in L^1\), we have \(\eta (z)\frac{ e^{{i\sigma z}}}{ i\sin {\sigma z}}\in L^1 (\mathbb R+i)\). Therefore \(\mu _{11}(z)e^{-i\sigma z}\in W_{\sigma }^{1}\). Similarly, we can prove that \(\mu _{21}(z)e^{i\sigma z}\in W_\sigma ^1\). As a consequence \(\mu _{11}\in E^1[\mathbb C(0;\pi /2)]\) and \(\mu _{21}\in E^1[\mathbb C(-\pi /2;0)]\).

Hishchak proves [22] that for \(f\in W^1_\sigma \) defined by (2.6) the decomposition

$$\begin{aligned} f=\mu _1-\mu _2,\quad \mu _1\in E^1[\mathbb C(0;\pi /2)], \quad \mu _2\in E^1[\mathbb C(-\pi /2;0)] \end{aligned}$$

is valid if and only if both of the following conditions are fulfilled

$$\begin{aligned} \sum \limits _{m=1}^{+\infty }\left| \sum \limits _{k=-\infty }^{+\infty }c_k\frac{k}{(m-\frac{i}{2}-k)(m-\frac{i}{2}-ik)}\right| <+\infty , \end{aligned}$$
(3.5)
$$\begin{aligned} \sum \limits _{m=1}^{+\infty }\left| \sum \limits _{k=-\infty }^{+\infty }c_k\frac{k}{(m+\frac{i}{2}+ik)(m+\frac{i}{2}-k)}\right| <+\infty . \end{aligned}$$
(3.6)

Note that \(\psi \) has representation (2.6). Let us consider condition (3.5) for \(\psi \) and denote by L the left hand side.

$$\begin{aligned} L=\frac{1}{\sqrt{2}}\sum \limits _{m=1}^{+\infty }\left| \sum \limits _{n=-\infty }^{+\infty }\frac{c_{2n}}{m-\frac{i}{2}-2n}-\frac{c_{2n}}{m-\frac{i}{2}-2ni} \right. \\ \left. +\sum \limits _{n=-\infty }^{+\infty }\frac{c_{2n+1}}{m-\frac{i}{2}-2n-1}-\frac{c_{2n+1}}{m-\frac{i}{2}-2ni-i}\right| . \end{aligned}$$

Since \(c_{2n}=-c_{2n+1}\), we obtain

$$\begin{aligned} L&=\frac{1}{\sqrt{2}}\sum \limits _{m=1}^{+\infty }\left| \sum \limits _{n=-\infty }^{+\infty }\frac{c_{2n}}{(m-\frac{i}{2}-2n)(m-\frac{i}{2}-2n-1)}\right. +\left. \frac{c_{2n}i}{(m-\frac{i}{2}-2ni)(m-\frac{i}{2}-2ni-i)}\right| \\&\quad \le \frac{1}{\sqrt{2}}\sum \limits _{m=1}^{+\infty }\left( \sum \limits _{n=-\infty }^{+\infty } \frac{|c_{2n}|}{|m-\frac{i}{2}-2n||m-\frac{i}{2}-2n-1|}+\sum \limits _{n=-\infty }^{+\infty }\frac{|c_{2n}|}{(m-\frac{i}{2}-2ni)(m-\frac{i}{2}-2ni-i)}\right) . \end{aligned}$$

Replacing \(l=m-2n\), we get

$$\begin{aligned} L\le & {} \frac{1}{\sqrt{2}}\sum \limits _{n=-\infty }^{+\infty }|c_{2n}|\sum \limits _{m=-\infty }^{+\infty }\frac{1}{\sqrt{(m-2n)^2+\frac{1}{4}}\sqrt{(m-2n-1)^2+\frac{1}{4}}}\\&+\frac{1}{\sqrt{2}}\sum \limits _{n=-\infty }^{+\infty }|c_{2n}|\sum \limits _{m=-\infty }^{+\infty }\frac{1}{\sqrt{m^2+(\frac{1}{2}+2n)^2}\sqrt{m^2+(\frac{1}{2}+2n+1)^2}}\\\le & {} \frac{1}{\sqrt{2}}\sum \limits _{n=-\infty }^{+\infty }|c_{2n}|\sum \limits _{l=-\infty }^{+\infty }\frac{1}{\sqrt{l^2+\frac{1}{4}}\sqrt{(l-1)^2+\frac{1}{4}}}\\&+\frac{1}{\sqrt{2}}\sum \limits _{n=-\infty }^{+\infty }|c_{2n}|\sum \limits _{m=-\infty }^{+\infty }\frac{1}{m^2}<+\infty \end{aligned}$$

Similarly we can prove (3.6) for \(\psi \).

The last step. We claim \(P_1\in E^1[\mathbb C(0;\pi /2)]\), \(P_1(z)e^{-2i\sigma z}\in E^1[\mathbb C(-\pi /2;0)]\).

It is sufficient to prove \(S\in E^1[\mathbb C(0;\pi /2)]\), \(S\in E^1[\mathbb C(-\pi /2;0)]\). Since \(\Phi _2=\beta _1-\beta _2\), \(\beta _1\in E^1[\mathbb C(0;\pi /2)]\), \(\beta _2\in E^1[\mathbb C(-\pi /2;0)]\), we get

Hence

$$\begin{aligned} S(z)=-\frac{1}{2\pi i}\int \limits _0^{+i\infty } {\frac{\Phi _1 (w)+\beta _1(w)}{w-z}dw} +\frac{1}{2\pi i}\int \limits _{-i\infty }^0 {\frac{\Phi _3 (w)+\beta _1(w)}{w-z}} dw. \end{aligned}$$

Then for \(z\in \mathbb C_+\) by (3.2) we have

$$\begin{aligned} S(z)=S(z)-S(-\overline{z})= \frac{1}{\pi }\int \limits _0^{\infty }\frac{x(\Phi _1 (iv)+\beta _1(iv))}{(v-y)^2+x^2}dv-\frac{1}{\pi }\int \limits _{-\infty }^0 \frac{x(\Phi _3 (iv)+\beta _2(iv))}{(v-y)^2+x^2}dv. \end{aligned}$$

Using Fubini’s theorem, this gives

$$\begin{aligned} \int \limits _{-\infty }^{+\infty }|S(x+iy)|dy\le \frac{1}{\pi } \int \limits _0^{\infty }|\Phi _1 (iv)+\beta _1(iv)|dv \int \limits _{-\infty }^{+\infty } \frac{x}{(v-y)^2+x^2}dy\\ +\frac{1}{\pi } \int \limits _{-\infty }^0|\Phi _3 (iv)+\beta _2(iv)|dv \int \limits _{-\infty }^{+\infty } \frac{x}{(v-y)^2+x^2}dy\le \alpha _1<+\infty , \end{aligned}$$

where \(\alpha _1\) does not depend on x. Similarly, \(\int \limits _{0}^{+\infty }|S(x+iy)|dx\le \alpha _2<+\infty ,\) where \(\alpha _2\) does not depend on y. Therefore (see [29]) \(S\in E^1[\mathbb C(0;\pi /2)]\), \(S\in E^1[\mathbb C(-\pi /2;0)]\).\(\square \)

The above proof motivates the following open problem.

Problem. Let \({\widehat{H}}^1_{2\sigma }=\left\{ f=f_1f_2:f_1\in {H}^2_{\sigma }, f_2\in {H}^2_{\sigma }\right\} \). It is clear that \({\widehat{H}}^1_{2\sigma }\subset {H}^1_{2\sigma }.\) But what about \({\widehat{H}}^1_{2\sigma }= {H}^1_{2\sigma }?\)

This problem is similar in some sense to the Ehrenpreis factorization problem (see [30, 31]).

If the answer is negative, the following question about the so-called weak factorization is interesting: is it possible for every function from \(H^1_{2\sigma }\) a representation as a sum (finite or infinite norm-controlled) of products of pairs of functions from \(H^2_\sigma \)?