Solvability Criterion for Convolution Equations on a Half-Strip

We obtain the criterion of solvability of homogeneous convolution equation in a half-strip. Proof is based on a new decomposition property of the weighted Hardy space. This result has relations to the spectral analysis-synthesis problem, cyclicity problem, information theory. All data generated or analysed during this study are included in this published article.


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 where unknown function f belongs to L 2 (−∞; 0). For the analysis of this equation the following characteristic functions 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 (C + ), 1 ≤ p < +∞, be the Hardy space of analytic in the half-plane C + := {z : z > 0} functions for which Properties of these spaces are described in details in [18]. Each function ψ ∈ H p (C + ), 1 ≤ p < +∞, has nontangential boundary values almost everywhere on the imaginary axis iR = ∂C + . These values we also define by ψ and ψ ∈ L p (iR).
N. Wiener and R. Paley [23] proved that (1.2) is a bijection of Let us define the function (iy) = F(iy)G(iy) almost everywhere on iR, 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). 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]).

The Main Result
First we need some definitions of analogues of the space L 2 (−∞; 0). Let Sedletskii [17] proved that the space H p (C + ) can be defined as the class of analytic on C + functions for which Therefore Vynnytskyi [20] has considered the following generalization of the Hardy space. Let H p σ (C + ), σ ≥ 0, 1 ≤ p < +∞, be the space of analytic in C + functions for which In [21] the following generalization of equation (1.1) Hence for the limit case σ = 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 and We define the Paley -Wiener space W p σ as the space of entire functions, whose restriction on R belongs to L p (R) and which satisfy the inequality | f (z)| < ce σ |z| for a positive constant C. Let iσ ], and l 3 = {w = u + iσ : −∞ < u ≤ 0}; l j are oriented anticlockwise with respect to D σ .

Lemma 2.1 [21] Equalities
Each function f ∈ W 2 σ can be represented as cardinal series by Whittaker-Kotelnikov-Shannon-Nyquist formula: ([12,14,15]) The space W 2 σ coincides with the space of functions represented as The space W 1 σ coincides with the space of functions represented as (2.6) such that The space W p σ can be defined also [16] as the space of entire functions satisfying the condition It is clear that H p σ (C + ) is a weighted Hardy space and is an analogue of the Paley-Wiener space for the half-plane as well. Let The main result of this paper is the following. (2.2) if and only if there exists a function P 1 , P 1 e −iσ z ∈ H 1 σ (C + ), such that nontangential boundary values of P 1 equal to 1 (it), t ∈ R, almost everywhere on the imaginary axis.

Proof of the Main Result
Sufficiency is proved in [25]. The proof of necessity is based on the following convolution theorem.
, then for all τ ≤ 0 the following equality The main idea of the following proof is the reduction of properties of the weighted Hardy space to the classical Hardy space.
Functions f ∈ E p [(α; γ )] have almost everywhere on ∂C(α; γ ) [9] the angular boundary values, which we denote by f and f ∈ L p [C(α; γ )]. The first step. We claim the function is the analytic continuation of 1 from iR to C + . We designate by η(τ ) the right hand side of (3.1). Since du.
The first integral in the left hand side of the above equality is the Poisson integral for the function ψ 1 (y) := 1 (iy), y > 0, 0, y < 0, therefore the nontangential boundary values from C + of this integral are equal to ψ 1 (y) almost everywhere on iR. Analogously the nontangential boundary values from C + of the second integral are equal to ψ 3 (y) := 3 (iy), y < 0, 0, y > 0, almost everywhere on iR. The third integral tends to zero as x tends to zero. Hence the nontangential boundary values of S(z) are equal to ψ(y) := 1 (iy), y > 0, − 3 (iy), y < 0.
It remains to prove P 1 (z)e −iσ z ∈ H 1 σ (C + ). The second step. There exist analytic in C + functions β 1 , β 2 such that 2 = β 1 −β 2 , It is easy to see By Paley-Wiener Theorem [18]  The third step. There exist analytic in C + functions μ 1 , μ 2 such that 22 Using Hölder's inequality, it is easy to see that 22 ∈ W 1 2σ . By Lemma 2.2 we have 22 First of all we prove that ψ ∈ W 1 2σ . Since series in (3.3) converges absolutely and uniformly on each compact set in C + , we have Each series above converges absolutely and uniformly on each compact set in C because sequence (b 2n + b 2n+1 ) belongs to l 1 . Therefore both the functions μ 11 and μ 21 are entire functions of exponential type ≤ σ . Let us remark that Since on the line z = x + i the function e iσ z i sin σ z is bounded and η ∈ L 1 , we have η(z) e iσ z i sin σ z ∈ L 1 (R + i). Therefore μ 11 (z)e −iσ z ∈ W 1 σ . Similarly, we can prove that μ 21 (z)e iσ z ∈ W 1 σ . As a consequence μ 11 ∈ E 1 [C(0; π/2)] and μ 21 ∈ E 1 [C(−π/2; 0)].
Hishchak proves [22] that for f ∈ W 1 σ defined by (2.6) the decomposition is valid if and only if both of the following conditions are fulfilled Note that ψ has representation (2.6). Let us consider condition (3.5) for ψ and denote by L the left hand side.
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 2σ a representation as a sum (finite or infinite norm-controlled) of products of pairs of functions from H 2 σ ?
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