Abstract
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.
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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
where unknown function f belongs to \(L^2 (-\infty ; 0)\). 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 \(H^p (\mathbb C _ +), 1 \le p <+ \infty , \) be the Hardy space of analytic in the half-plane \(\mathbb C _ +: = \{z: \mathfrak {R}z> 0 \} \) functions for which
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
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
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
In [21] the following generalization of equation (1.1)
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
and
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
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
The space \(W_\sigma ^1\) coincides with the space of functions represented as (2.6) such that \((c_k)\in l^1,\)
The space \(W_\sigma ^p\) can be defined also [16] as the space of entire functions satisfying the condition
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
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
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
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
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
by Fubini theorem we have
Hence for \(z\in \mathbb C_+\)
The first integral in the left hand side of the above equality is the Poisson integral for the function
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
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
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
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
and
Therefore
where
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
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
where
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
Since on the line \(z=x+i\) the function
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
is valid if and only if both of the following conditions are fulfilled
Note that \(\psi \) has representation (2.6). Let us consider condition (3.5) for \(\psi \) and denote by L the left hand side.
Since \(c_{2n}=-c_{2n+1}\), we obtain
Replacing \(l=m-2n\), we get
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
![](http://media.springernature.com/lw440/springer-static/image/art%3A10.1007%2Fs11785-021-01120-8/MediaObjects/11785_2021_1120_Equ45_HTML.png)
Hence
Then for \(z\in \mathbb C_+\) by (3.2) we have
Using Fubini’s theorem, this gives
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 \(H^2_\sigma \)?
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Dilnyi, V. Solvability Criterion for Convolution Equations on a Half-Strip. Complex Anal. Oper. Theory 15, 73 (2021). https://doi.org/10.1007/s11785-021-01120-8
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DOI: https://doi.org/10.1007/s11785-021-01120-8
Keywords
- Convolution equation
- Hardy space
- Analytic continuation
- Spectral analysis
- Spectral synthesis
- Signal processing