δ-β-Gabor integral operators for a space of locally integrable generalized functions

Abstract

In this article, we give a definition and discuss several properties of the δ-β-Gabor integral operator in a class of locally integrable Boehmians. We derive delta sequences, convolution products and establish a convolution theorem for the given δ-β-integral. By treating the delta sequences, we derive the necessary axioms to elevate the δ-β-Gabor integrable spaces of Boehmians. The said generalized δ-β-Gabor integral is, therefore, considered as a one-to-one and onto mapping continuous with respect to the usual convergence of the demonstrated spaces. In addition to certain obtained inversion formula, some consistency results are also given.

1 Preliminaries

Due to various applications of the Dirac delta distribution and its usual implementation in physics, engineering, and partial differential equations, various distribution spaces have been considered in literature. Distributions or generalized functions are objects that generalize the classical notion of functions. They are widely used in geometry, mathematical physics, stochastic analysis, harmonic analysis, and the theory of linear and nonlinear partial differential equations as it is much easier to establish a distributional solution than a classical one. The space of Boehmians is a space of generalized functions constructed in an algebraic way similar to the construction of the field of quotients. When a multiplication is interpreted as a convolution, the construction of a space of Boehmians, applied to a different function space, yields a different space of Boehmians. Boehmians also allow different identifications of integral operators to be isomorphisms. Therefore, several integral operators have been applied to various spaces of Boehmians in various papers in the recent past. For example, the Stieltjes integral operator was extended to a space of Boehmians of Fox’s H-function type (see, e.g., [1]), the Hartley integral operator was extended to a space of strong Boehmians (see, e.g., [2]), the Hilbert integral operator was extended to a space of Boehmians (see, e.g., [3]), the Mellin integral operator was extended to a space of Boehmians of quaternion type (see, e.g., [4]), the short-time Fourier integral operator was extended to a space of H-Boehmians (see, e.g., [5]), the fractional Fourier integral was extended to a space of Boehmians (see, e.g., [6]), the quaternion Fourier integral was extended to a quaternionic set of Boehmians (see, e.g., [7]) and many others, to mention but a few. However, further extensions of integral operators to Boehmian spaces were obtained in [1–10] (see also citations therein).

The Gabor integral transform operator, among other integral transform operators, was proposed as a time-frequency integral operator to perform simultaneous time-frequency analysis of signals. This frequently was used for feature extraction, non-stationary signal processing, radar systems, sonar systems, communications, and space sciences (see, e.g., [11]). For a given window function g and coordinates in the space and the frequency domains q and q̃, the Gabor integral operator of a signal φ is given as follows (see, e.g., [12]):

$$ G_{g}\varphi ( q,\tilde{q} ) = \int _{\mathbb{R}}\varphi ( x ) \overline{g ( x-\tilde{q} ) }\exp ( i2 \pi qx ) \,dx, $$

(1)

when the integral exists. If g is a given window function satisfying the integral equation

$$ \int _{\mathbb{R}} \bigl\vert g ( x ) \bigr\vert ^{2}\,dx=1, $$

then the signal function φ can be recovered from the Gabor spectrum integral $G_{g}\varphi$ as follows:

$$ \varphi ( x ) = \int _{\mathbb{R}} \int _{\mathbb{R}}G_{g} \varphi ( q,\tilde{q} ) g ( x-q ) \overline{\exp ( i2\pi qx ) }\,dq \,d\tilde{q}. $$

(2)

The Gabor integral operator of an image has much lower entropy than the pixel representation of the image, and the expansions of the Gabor, for every low bit rates, can provide better signal compression than the discrete cosine integral operator can (see, e.g., [8, 10–18]). If δ and β are real numbers, when β is fixed, then the collection $g_{q,\tilde{q}}^{\beta ,\delta }$ of functions is defined by

$$ g_{q,\tilde{q}}^{\beta ,\delta } ( x ) =\exp \biggl( \frac{2iqx-q\tilde{q}}{2} \biggr) g_{\delta }^{\beta } ( x-\tilde{q} ), $$

(3)

where $g_{\delta }^{\beta }$ is a family of Gaussian window functions defined by

$$ g_{\delta }^{\beta } ( x ) =\delta ^{\frac{1}{4}}\exp \biggl( \frac{i\beta x^{2}-2\delta x^{2}}{2} \biggr) . $$

(4)

With respect to the family $g_{\delta }^{\beta }$ of window functions, the δ-β-Gabor integral operator of a signal φ is defined by (see, e.g., [13, Eq. (4)])

$$ G_{\delta }^{\beta }\varphi ( q,\tilde{q} ) = \int _{ \mathbb{R}}g_{q,\tilde{q}}^{\beta ,\delta } ( x ) \varphi ( x ) \,dx. $$

(5)

The δ-β-Gabor integral operator is the δ-β-extension of the Gabor integral operator when $\delta =1$ and $\beta =0$. It is closely related to the Wigner–Ville integral operator

$$ W \bigl( g_{\delta }^{\beta } \bigr) ( x,\xi ) =C\exp \biggl( -\frac{2\delta +\beta ^{2}}{2\delta }x^{2}-\frac{1}{2\delta }\xi ^{2}+ \frac{\beta }{\delta }x\xi \biggr) , $$

and they are related by (see, e.g., [13])

$$ \bigl\vert G_{\delta }^{\beta }\varphi ( q,\tilde{q} ) \bigr\vert ^{2}=W \bigl( g_{\delta }^{\beta } \bigr) \sharp W ( \varphi ), $$

where ♯ is the convolution product of the Fourier convolution type (see, e.g., [19])

$$ \varphi \sharp \psi ( \xi ) = \int _{\mathbb{R}}\psi ( t ) \varphi ( \xi -t ) \,dt, $$

(6)

and C is some constant. However, the δ-β-Gabor integral of a signal may look much better than the Gabor integral operator since the choice of δ and β, which brings out features best, will depend on the time-frequency content of the signal itself. It, therefore, becomes natural to adapt the choice of parameters δ and β to the phase point $( q,\tilde{q} )$. However, this article firstly aims to discuss convolution products and convolution theorems for the δ-β-Gabor integral operator. It then generates two sets of Boehmians and gives some characteristics of the extended δ-β-Gabor integral operator. For the convenience of the reader, we distribute our results into four sections. In Sect. 2, we introduce convolution products and prove a convolution theorem. In Sect. 3, we generate the δ-β-Gabor sets of Boehmians. In Sect. 4, we discuss the δ-β-Gabor integral on the set of Boehmians and obtain an inversion formula as well as radical properties of the generalized integral.

2 Convolutions and convolution theorem

In mathematics, the convolution is a mathematical operation on two functions, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as one of the original functions is translated. Convolutions have applications including probability, statistics, computer vision, image and signal processing, electrical engineering, and differential equations. At any rate, the purpose of this section is to devise the convolution product and derive the convolution theorem for the δ-β-Gabor integral operator for arbitrary real numbers δ and β.

The analysis of the Gabor integral operator successfully brings light to a very beneficial convolution product that works in with the Fourier convolution product. The compatible convolution product is defined as follows.

Definition 1

Let $\varphi \in L^{1} ( \mathbb{R} )$ and $F\in L^{1} ( \mathbb{R}^{2} )$. Then, for φ and F, we define an integral equation $\star _{q}^{\tilde{q}}$ as follows:

$$ \varphi \star _{q}^{\tilde{q}}F ( q,\tilde{q} ) = \int _{ \mathbb{R}}\varphi ( x ) \exp \biggl( \frac{2iqx-qx}{2} \biggr) F ( q,\tilde{q}-x ) \,dx, $$

(7)

provided the right-hand side integral exists for all coordinates q and q̃.

To establish the δ-β-Gabor convolution theorem, we firstly derive the following preliminary result.

Theorem 2

Let $g_{q,\tilde{q}}^{\beta ,\delta }$and $g_{\delta }^{\beta }$be defined as in Eq. (3) and Eq. (4), respectively. Then, for real numbers z and x, we have

$$ g_{q,\tilde{q}}^{\beta ,\delta } ( z+x ) =\exp \biggl( \frac{2iqx-qx}{2} \biggr) \exp \biggl( \frac{2iqz-q ( \tilde{q}-t ) }{2} \biggr) g_{\delta }^{ \beta } ( z ) . $$

Proof

By using Eq. (3) and employing the change of variables $w=z+x-\tilde{q}$, the above equation routinely becomes

$$ g_{q,\tilde{q}}^{\beta ,\delta } ( z+x ) =\exp \biggl( \frac{2i ( q+z ) -q\tilde{q}}{2} \biggr) g_{\delta }^{\beta } ( w ). $$

Hence, by taking into account the definition of a family of the Gaussian window functions $g_{\delta }^{\beta }$, we get

$$ g_{q,\tilde{q}}^{\beta ,\delta } ( z+x ) =\exp \bigl( iq ( z+x ) \bigr) \exp \biggl( -\frac{q\tilde{q}}{2} \biggr) \delta ^{\frac{1}{4}}\exp \biggl( \frac{i\beta w^{2}-2\delta w^{2}}{2} \biggr). $$

This can alternatively be written as

$$ g_{q,\tilde{q}}^{\beta ,\delta } ( z+x ) =\exp ( ipx ) \exp \biggl( \frac{2ipz-q ( \tilde{q}-x ) -qx}{2} \biggr) \delta ^{\frac{1}{4}}\exp \biggl( \frac{i\beta w^{2}-2\delta w^{2}}{2} \biggr) . $$

Therefore, by using a simple computation and making a rearrangement on the above exponents yield

$$ g_{q,\tilde{q}}^{\beta ,\delta } ( z+x ) =\exp \biggl( \frac{2iq-qx}{2} \biggr) \exp \biggl( \frac{2iqz-q ( \tilde{q}-x ) }{2} \biggr) \delta ^{\frac{1}{4}}\exp \biggl( \frac{i\beta w^{2}-2\delta w^{2}}{2} \biggr). $$

Indeed, the definition of $g_{\delta }^{\beta }$ gives

$$ g_{q,\tilde{q}}^{\beta ,\delta } ( z+x ) =\exp \biggl( \frac{2iqx-qx}{2} \biggr) \exp \biggl( \frac{2iqz-q ( \tilde{q}-x ) }{2} \biggr) g_{\delta }^{ \beta } ( z ) . $$

This finishes the proof of the theorem. □

Now, as we are implementing the Fourier convolution product ♯ in our next investigation, we have to be very familiar with the convolution properties on the set of integrable functions that we recall (see, e.g., [6]):

$$ \varphi \sharp \psi =\psi \sharp \varphi \quad \text{and}\quad \varphi \sharp ( \psi \sharp \theta ) = ( \psi \sharp \varphi ) \sharp \theta . $$

On the basis of the above definitions, the convolution theorem of the Gabor integral $G_{\delta }^{\beta }$ can be derived as follows.

Theorem 3

Let φ and ψ be arbitrarily given in $L^{1} ( \mathbb{R} )$, and let q and q̃ be the coordinates in the space and frequency domains, then we have

$$ G_{\delta }^{\beta } ( \varphi \sharp \psi ) ( q,\tilde{q} ) = \bigl( G_{\delta }^{\beta }\varphi \star _{q}^{ \tilde{q}} \psi \bigr) ( q,\tilde{q} ) . $$

Proof

Let φ and ψ be arbitrarily given. Then, by considering the integral relation presented in Eq. (7), we write

$$ G_{\delta }^{\beta } ( \varphi \sharp \psi ) ( q,\tilde{q} ) = \int _{\mathbb{R}}\psi ( t ) \int _{ \mathbb{R}}g_{q,\tilde{q}}^{\beta ,\delta } ( x ) \varphi ( x-t ) \,dx \,dt. $$

(8)

Hence, the substitution $x=z+t$, consequently, changes Eq. (8) into the integral relation

$$ G_{\delta }^{\beta } ( \varphi \sharp \psi ) ( q,\tilde{q} ) = \int _{\mathbb{R}}\psi ( t ) \int _{ \mathbb{R}}g_{q,\tilde{q}}^{\beta ,\delta } ( z+t ) \varphi ( z ) \,dz \,dt. $$

(9)

Thus, with the aid of Theorem 2, Eq. (9) can be nicely expressed as

$$ G_{\delta }^{\beta } ( \varphi \sharp \psi ) ( q,\tilde{q} ) = \int _{\mathbb{R}}\psi ( t ) \exp \biggl( \frac{2iqt-qt}{2} \biggr) \biggl( \int _{\mathbb{R}}g_{q,\tilde{q}-t}^{\beta ,\delta } ( z ) \varphi ( z ) \,dz \biggr) \,dt. $$

Therefore, by employing Eq. (5), the Gabor integral of the convolution can be given as follows:

$$ G_{\delta }^{\beta } ( \varphi \sharp \psi ) ( q,\tilde{q} ) = \int _{\mathbb{R}}\psi ( t ) \exp \biggl( \frac{2iqt-qt}{2} \biggr) G_{\delta }^{\beta } ( \varphi ) ( q, \tilde{q}-t ) \,dt. $$

Hence, by using Eq. (7) we complete the proof of the theorem. □

Indeed, analogous to many magnificent research works related to various integral operators, this convolution theorem still lacks the elegance and simplicity of the Euclidean Fourier integral operator, which states that the Fourier integral operator of a convolution of two functions is the product of their respective Fourier transforms. Although the convolution product and the convolution theorem of the δ-β-Gabor integral operator are not so accessible as the Euclidean Fourier convolution product and convolution theorem, they will serve us nicely to present a simple though rigorous approach to the generalized δ-β-Gabor spaces of Boehmians. It also serves to derive a linear bijection between the Boehmian spaces.

3 δ-β-Gabor spaces of Boehmians

Denote by $D ( \mathbb{R} )$ the space of smooth functions of compact supports over $\mathbb{R}$ and, at the same time, denote by $L_{loc}^{1} ( \mathbb{R}^{2} )$ the space of locally integrable functions over $\mathbb{R}^{2}$. Denote by Δ the set of delta sequences $\{ \delta _{n} \}$ from $( D ( \mathbb{R} ) ,\sharp )$ satisfying $\Delta _{1}-\Delta _{3}$, where

$$\begin{aligned}& \Delta _{1}: \int _{\mathbb{R}}\delta _{n}=1 \text{ for all } n\in \mathbb{N}. \\& \Delta _{2}: \vert \delta _{n} \vert < M, M\in \mathbb{R}, 0< M , n\in \mathbb{N}. \\& \Delta _{3}:supp\delta _{n}\subseteq ( -\gamma _{n},\gamma _{n} ) ,\gamma _{n}\rightarrow 0 \text{ as } n\rightarrow \infty . \end{aligned}$$

With $L_{loc}^{1} (\mathbb{R}^{2} )$, $D ( \mathbb{R} )$, Δ, the product ♯, and the product $\star _{q}^{\tilde{q}}$, we generate a space $B ( \mathbb{R}^{2} )$ of Boehmians which works in as a range space of the generalized δ-β-Gabor integral operator. Therefore, we start with the proof of the following fundamental result.

Lemma 4

Let $F\in L_{loc}^{1} ( \mathbb{R}^{2} )$and $\varphi ,\psi \in D ( \mathbb{R} )$. Then we have

$$ F\star _{q}^{\tilde{q}} ( \varphi \sharp \psi ) ( q,\tilde{q} ) = \bigl( F\star _{q}^{\tilde{q}}\varphi \bigr) \star _{q}^{\tilde{q}}\psi ( q,\tilde{q} )\quad \textit{in }L_{loc}^{1} \bigl( \mathbb{R}^{2} \bigr), $$

where $( q,\tilde{q} ) \in \mathbb{R}^{2}$.

Proof

By employing the definitions of the convolution products $\star _{q}^{\tilde{q}}$ and ♯, we respectively obtain

$$\begin{aligned} F\star _{q}^{\tilde{q}} ( \varphi \sharp \psi ) ( q,\tilde{q} ) =& \int _{\mathbb{R}}F ( q,\tilde{q}-x ) \exp \biggl( \frac{2iqx-qx}{2} \biggr) ( \varphi \sharp \psi ) ( x ) \,dx \\ =& \int _{\mathbb{R}}F ( q,\tilde{q}-x ) \exp \biggl( \frac{2iqx-qx}{2} \biggr) \int _{\mathbb{R}}\varphi ( x-y ) \psi ( y ) \,dy \,dx. \end{aligned}$$

Hence, with the aid of Fubini’s theorem, we simplify the above equation to write

$$ F\star _{q}^{\tilde{q}} ( \varphi \sharp \psi ) ( q,\tilde{q} )= \int _{\mathbb{R}}\psi ( y ) \int _{ \mathbb{R}}F ( q,\tilde{q}-x ) \exp \biggl( \frac{2iqx-qx}{2} \biggr) \varphi ( x-y ) \,dx \,dy. $$

(10)

Therefore, a proper change in the variables changes Eq. (10) into the standard form

$$\begin{aligned} &F\star _{q}^{\tilde{q}} ( \varphi \sharp \psi ) ( q,\tilde{q} ) \\ &\quad = \int _{\mathbb{R}}\psi ( y ) \exp \biggl( \frac{2iqy-qy}{2} \biggr) \int _{\mathbb{R}}F \bigl( q ( \tilde{q}-y ) -z \bigr) \exp \biggl( \frac{2iqx-qx}{2} \biggr) \varphi ( z ) \,dz \,dy \\ &\quad = \int _{\mathbb{R}}\psi ( y ) \exp \biggl( \frac{2iqy-qy}{2} \biggr) \bigl( F\star _{q}^{\tilde{q}} \varphi \bigr) ( q,\tilde{q}-y ) \,dy. \end{aligned}$$

Hence, we have reached the conclusion that $F\star _{q}^{\tilde{q}} ( \varphi \sharp \psi ) = ( F \star _{q}^{\tilde{q}}\varphi ) \star _{q}^{\tilde{q}}\psi$. To complete the proof of this lemma, we have to show

$$ F\star _{q}^{\tilde{q}}\varphi \in L_{loc}^{1} \bigl( \mathbb{R}^{2} \bigr) $$

(11)

for every $F\in L_{loc}^{1} ( \mathbb{R}^{2} )$ and $\phi \in D ( \mathbb{R} )$. Let K be a compact subset of $R^{2}$, then we have

$$\begin{aligned} \int _{K} \bigl\vert \bigl( F\star _{q}^{\tilde{q}} \varphi \bigr) ( q,\tilde{q} ) \bigr\vert \,d ( q,\tilde{q} ) =& \int _{K} \biggl\vert \int _{\mathbb{R}}F ( q,\tilde{q}-x ) \varphi ( x ) \exp \biggl( \frac{2iqx-qx}{2} \biggr) \,dx \biggr\vert \,d ( q,\tilde{q} ) \\ \leq & \int _{\mathbb{R}} \bigl\vert \varphi ( x ) \bigr\vert \int _{K} \bigl\vert F ( q,\tilde{q}-x ) \bigr\vert \,d ( q, \tilde{q} ) \,dx \\ \leq &A \int _{\mathbb{R}} \bigl\vert \varphi ( x ) \bigr\vert \,dx, \end{aligned}$$

where A is a certain positive constant such that

$$ \int _{K} \bigl\vert F ( q,\tilde{q}-x ) \bigr\vert \,d ( q, \tilde{q} ) \leq A\text{ as }F\in L_{loc}^{1} \bigl( \mathbb{R}^{2} \bigr). $$

(12)

Also, as $\varphi \in D ( \mathbb{R} )$ and φ is a smooth function of compact support, we have

$$ \int _{\mathbb{R}} \bigl\vert \varphi ( x ) \bigr\vert \,dx \leq B $$

for some $B\in \mathbb{R}$. Hence, we have obtained $F\star _{q}^{\tilde{q}}\varphi \in L_{loc}^{1} ( \mathbb{R}^{2} )$ for all $F\in L_{loc}^{1} ( \mathbb{R}^{2} )$ and $\varphi \in D ( \mathbb{R} )$.

The proof of the theorem is, therefore, finished. □

Each of the identities of the following lemma has a routine verification and, hence, the proof is left to the reader. Details are, therefore, omitted.

Lemma 5

Let $F,\{F_{n}\}\in L_{loc}^{1} ( \mathbb{R}^{2} )$and $\varphi ,\psi \in D ( \mathbb{R} )$and $\alpha \in \mathbb{C}$. Then we have $F\star _{q}^{\tilde{q}} ( \varphi +\psi ) =F\star _{q}^{ \tilde{q}}\varphi +F\star _{q}^{\tilde{q}}\psi$, $\text{ } ( \alpha F ) \star _{q}^{\tilde{q}}\varphi =\alpha ( F \star _{q}^{\tilde{q}}\varphi )$and

$$ F_{n}\star _{q}^{\tilde{q}}\varphi \rightarrow F\star _{q}^{\tilde{q}} \varphi\quad \textit{as }n\rightarrow \infty \textit{ as } F_{n}\rightarrow F\textit{ as }n\rightarrow \infty .$$

Lemma 6

For $\{\delta _{n}\}\in \Delta$and $F\in L_{loc}^{1} ( \mathbb{R}^{2} )$, we have

$$ F\star _{q}^{\tilde{q}}\delta _{n}\rightarrow F\quad \textit{in }L_{loc}^{1} \bigl( \mathbb{R}^{2} \bigr) \textit{ as }n\rightarrow \infty . $$

(13)

Proof

Let $K\subseteq \mathbb{R}^{2}$ be compact. Then, by using the concept of $\Delta _{1}$, we have

$$\begin{aligned} &\int _{K} \bigl\vert \bigl( F\star _{q}^{\tilde{q}} \delta _{n}-F \bigr) ( q,\tilde{q} ) \bigr\vert \,d ( q, \tilde{q} ) \\ &\quad = \int _{K} \biggl\vert \bigl( F\star _{q}^{ \tilde{q}} \delta _{n} \bigr) ( q,\tilde{q} ) -F ( q, \tilde{q} ) \int _{\mathbb{R}}\delta _{n} ( x ) \,dx \biggr\vert \,d ( q, \tilde{q} ) \\ &\quad \leq \int _{K} \biggl( \int _{\mathbb{R}} \biggl\vert F ( q, \tilde{q}-x ) \exp \biggl( \frac{2iqx-qx}{2} \biggr) -F ( q, \tilde{q} ) \biggr\vert \bigl\vert \delta _{n} ( x ) \bigr\vert \,dx \biggr) \,d ( q,\tilde{q} ) \\ &\quad \leq \int _{K} \biggl( \int _{-\gamma _{n}}^{\gamma _{n}} \bigl\vert F ( q, \tilde{q}-x ) -F ( q,\tilde{q} ) \bigr\vert \bigl\vert \delta _{n} ( x ) \bigr\vert \,dx \biggr) \,d ( q,\tilde{q} ) \\ &\quad \leq M \int _{K} \int _{-\gamma _{n}}^{\gamma _{n}} \bigl\vert F ( q, \tilde{q}-x ) -F ( q,\tilde{q} ) \bigr\vert \,dx\,d ( q,\tilde{q} ) . \end{aligned}$$

The last inequality follows from $\Delta _{2}$ and the fact that $\{ \delta _{n} \} \subseteq D ( \mathbb{R} )$. Hence, as $F\in L_{loc}^{1} ( \mathbb{R}^{2} )$, by pursuing simple computations, we write

$$ \int _{K} \bigl\vert \bigl( F\star _{q}^{\tilde{q}} \delta _{n} \rightarrow F \bigr) ( q,\tilde{q} ) \bigr\vert \,d ( q, \tilde{q} ) \leq MA\mu ( K ) ( 2 \gamma _{n} ) , $$

where $\mu ( K )$ is the Lebesgue measure of K and A is some positive constant. Hence, by $\Delta _{3}$, we have

$$ \bigl\Vert F\star _{q}^{\tilde{q}}\delta _{n}-F \bigr\Vert \leq MA \mu ( K ) ( 2\gamma _{n} ) \rightarrow 0 $$

as $n\rightarrow \infty$. This finishes the proof of the theorem. □

The Boehmian space $B ( \mathbb{R}^{2} )$ with the sets $( L_{loc}^{1} ( \mathbb{R}^{2} ) ,\star _{q}^{ \tilde{q}} )$, $( D ( \mathbb{R} ) ,\sharp )$, $\Delta ( \mathbb{R} )$ is defined.

The sum of the Boehmians $\varphi _{n}/\delta _{n}$ and $g_{n}/\varepsilon _{n}$ in $B ( \mathbb{R}^{2} )$ is given as

$$ \varphi _{n}/\delta _{n}+g_{n}/\varepsilon _{n}= \bigl( \varphi _{n} \star _{q}^{\tilde{q}} \delta _{n}+g_{n}\star _{q}^{\tilde{q}}\delta _{n} \bigr) / ( \delta _{n}\sharp \varepsilon _{n} ) , $$

whereas a multiplication of a Boehmian $\varphi _{n}/\delta _{n}$ in $B ( \mathbb{R}^{2} )$ by a complex number $\gamma \in \mathbb{C}$ is defined as $\gamma ( \varphi _{n}/\delta _{n} ) = ( \gamma \varphi _{n}/\delta _{n} )$. On the other hand, the extension of $\star _{q}^{\tilde{q}}$ and $D^{\alpha }$ to $B ( \mathbb{R}^{2} )$ is introduced as follows:

$$ ( \varphi _{n}/\delta _{n} ) \star _{q}^{\tilde{q}} ( g_{n}/\varepsilon _{n} ) = \bigl( \varphi _{n} \star _{q}^{ \tilde{q}}g_{n} \bigr) / ( \delta _{n}\sharp \varepsilon _{n} )\quad \text{and}\quad D^{\alpha } ( \varphi _{n}/\delta _{n} ) = \bigl( D^{\alpha } \varphi _{n}/\delta _{n} \bigr) ,\alpha \in \mathbb{R}. $$

Moreover, an extension of $\star _{q}^{\tilde{q}}$ to $B ( \mathbb{R}^{2} ) \star _{q}^{\tilde{q}}L_{loc}^{1} ( \mathbb{R}^{2} )$, where $( \varphi _{n}/\delta _{n} )$ is in $B ( \mathbb{R}^{2} )$ and ω in $L_{loc}^{1} ( \mathbb{R}^{2} )$, is given as

$$ ( \varphi _{n}/\delta _{n} ) \star _{q}^{\tilde{q}} \omega = \bigl( \varphi _{n}\star _{q}^{\tilde{q}}\omega \bigr) / \delta _{n}. $$

Definition 7

Let $\beta _{n},\beta \in B ( \mathbb{R}^{2} )$ for $n=1,2,3,\ldots$ . Then the sequence $\{ \beta _{n} \}$ is δ-convergent to β, denoted by $\delta -\lim_{n\rightarrow \infty }\beta _{n}=\beta ( \beta _{n} \overset{\delta }{\rightarrow }\beta )$, provided there can be found a delta sequence $\{ \delta _{n} \}$ such that

(a) $( \beta _{n}\star _{q}^{\tilde{q}}\delta _{k} )$ and $( \beta \star _{q}^{\tilde{q}}\delta _{k} ) \in L_{ loc}^{1} ( \mathbb{R}^{2} )$ for all $n,k\in \mathbb{N}$,

(b) $\lim_{n\rightarrow \infty }\beta _{n}\star _{q}^{\tilde{q}}\delta _{k}=\beta \star _{q}^{\tilde{q}}\delta _{k}$ in $L_{loc} ^{1} ( \mathbb{R}^{2} )$ for every $k\in \mathbb{N}$.

Or, equivalently, $\delta -\lim_{n\rightarrow \infty }\beta _{n}=\beta$ if and only if there are $\varphi _{n,k}$, $\varphi _{k}\in L_{loc}^{1} ( \mathbb{R}^{2} )$ and $\{ \delta _{k} \} \in \Delta$ such that (i) $\beta _{n}=\varphi _{n,k}/\delta _{k}$, $\beta =\varphi _{k}/\delta _{k}$ (ii) $\lim_{n\rightarrow \infty }\varphi _{n,k}=\varphi _{k}\in L_{loc}^{1} ( \mathbb{R}^{2} )$ to every $k\in \mathbb{N}$.

Definition 8

Let $\beta _{n},\beta \in B ( \mathbb{R}^{2} )$ for $n=1,2,3,\ldots$ . Then the sequence $\{ \beta _{n} \}$ is Δ-convergent to β, denoted by Δ-$\lim_{n\rightarrow \infty }\beta _{n}=\beta ( \beta _{n} \overset{\Delta }{\rightarrow }\beta )$, provided there can be found a delta sequence $\{ \delta _{n} \}$ such that

(i) $( \beta _{n}-\beta ) \star _{q}^{\tilde{q}}\delta _{n}\in L_{loc}^{1} ( \mathbb{R}^{2} )$$( \forall n\in \mathbb{N} )$ (ii) $\lim_{n\rightarrow \infty } ( \beta _{n}- \beta ) \star _{q}^{\tilde{q}}\delta _{n}=0$ in $L_{loc} ^{1} ( \mathbb{R}^{2} )$.

Defining the space H with the sets $( L_{loc}^{1} ( \mathbb{R} ) ,\sharp )$, $( D ( \mathbb{R} ) ,\sharp )$, and $\Delta ( \mathbb{R} )$ is quite similar to the construction of the space of $L^{p}$- Boehmians; for details, we refer to [9]. In H, addition of $\varphi _{n}/\delta _{n}$ and $g_{n}/\varepsilon _{n}$ in H and ♯ is, respectively, defined as

$$\begin{aligned}& \varphi _{n}/\delta _{n}+g_{n}/\varepsilon _{n} = ( \varphi _{n} \sharp \delta _{n}+g_{n} \sharp \delta _{n} ) / ( \delta _{n} \sharp \varepsilon _{n} ) \\& \text{and} \\& ( \varphi _{n}/\delta _{n} ) \sharp ( g_{n}/ \varepsilon _{n} ) = ( \varphi _{n}\sharp g_{n} ) / ( \delta _{n}\sharp \varepsilon _{n} ) . \end{aligned}$$

Multiplication of $\varphi _{n}/\delta _{n}$ in H by a complex number $\gamma \in \mathbb{C}$ is defined as

$$ \gamma ( \varphi _{n}/\delta _{n} ) = ( \gamma \varphi _{n}/\delta _{n} ) . $$

$D^{\alpha }$ of $\varphi _{n}/\delta _{n}$ in H is introduced as $D^{\alpha } ( \varphi _{n}/\delta _{n} ) =D^{\alpha } \varphi _{n}/\delta _{n}$, $\alpha \in \mathbb{R}$. For every $\varphi _{n}/\delta _{n}$ in H and κ in $L_{loc}^{1} ( \mathbb{R} )$, ♯ can be extended to $H\sharp L_{loc}^{1} ( \mathbb{R} )$ by $( \varphi _{n}/\delta _{n} ) \sharp \kappa = ( \varphi _{n}\sharp \kappa ) /\delta _{n}$.

Definition 9

Let $\beta _{n},\beta \in H$ for $n=1,2,3,\ldots$ . Then the sequence $\{ \beta _{n} \}$ is δ-convergent to β, denoted by $\delta -\lim_{n\rightarrow \infty }\beta _{n}=\beta ( \beta _{n} \overset{\delta }{\rightarrow }\beta )$, provided there can be found a delta sequence $\{ \delta _{n} \}$ such that

(i) $\beta _{n}\sharp \delta _{k}$ and $\beta \sharp \delta _{k}\in L_{loc}^{1} ( \mathbb{R} )$ for all $n,k\in \mathbb{N}$,

(ii) $\lim_{n\rightarrow \infty }\beta _{n}\sharp \delta _{k}=\beta \sharp \delta _{k}$ in $L_{loc}^{1} ( \mathbb{R} )$ for every $k\in \mathbb{N}$.

Or, equivalently,

$\delta -\lim_{n\rightarrow \infty }\beta _{n}=\beta$ if and only if there are $\varphi _{n,k}$, $\varphi _{k}\in L_{loc}^{1} ( \mathbb{R} )$ and $\{ \delta _{k} \} \in \Delta$ such that

(i) $\beta _{n}=\varphi _{n,k}/\delta _{k}$, $\beta =\varphi _{k}/\delta _{k}$,

(ii) to every $k\in \mathbb{N}$, we have $\lim_{n\rightarrow \infty }\varphi _{n,k}=\varphi _{k}$ in $L_{loc}^{1} ( \mathbb{R} )$.

Definition 10

Let $\beta _{n},\beta \in H$ for $n=1,2,3,\ldots$ . Then the sequence $\{ \beta _{n} \}$ is Δ-convergent to β, denoted by Δ-$\lim_{n\rightarrow \infty }\beta _{n}=\beta ( \beta _{n} \overset{\Delta }{\rightarrow }\beta )$, provided there can be found a delta sequence $\{ \delta _{n} \}$ such that

(i) $( \beta _{n}-\beta )\sharp \delta _{n}\in L_{loc}^{1} ( \mathbb{R} )$$( \forall n\in \mathbb{N} )$,

(ii) $\lim_{n\rightarrow \infty } ( \beta _{n}- \beta ) \sharp \delta _{n}=0$ in $L_{loc}^{1} ( \mathbb{R} )$.

4 The $X_{\delta }^{\beta }$ and $S_{\delta }^{\beta }$ integrals

With the aid of the previous investigation, we estimate the Gabor $X_{\delta }^{\beta }$ integral and its inversion $S_{\delta }^{\beta }$ on the respective spaces $H ( \mathbb{R} )$ and $B ( \mathbb{R}^{2} )$ as follows.

Definition 11

Let $\varphi _{n}/\delta _{n}\in H ( \mathbb{R} )$, then the estimated δ-β-Gabor integral of $\varphi _{n}/\delta _{n}$ is defined as

$$ X_{\delta }^{\beta } ( \varphi _{n}/\delta _{n} ) =G_{ \delta }^{\beta }\varphi _{n}/\delta _{n}, $$

(14)

which lies in the space $B ( \mathbb{R}^{2} )$.

We recite some properties of the extension $X_{\delta }^{\beta }$ of $G_{\delta }^{\beta }$ with the help of the following theorems.

Theorem 12

(i) $X_{\delta }^{\beta }:H ( \mathbb{R} ) \rightarrow B ( \mathbb{R}^{2} )$is well defined.

(ii) $X_{\delta }^{\beta }:H ( \mathbb{R} ) \rightarrow B ( \mathbb{R}^{2} )$is linear, one-to-one, and onto.

(iii) $X_{\delta }^{\beta }:H ( \mathbb{R} ) \rightarrow B ( \mathbb{R}^{2} )$is continuous with respect to δ and Δ-convergence.

(iv) $X_{\delta }^{\beta }:H ( \mathbb{R} ) \rightarrow B ( \mathbb{R}^{2} )$is consistent with the operator $G_{\delta }^{\beta }$.

(v) $X_{\delta }^{\beta } ( ( \varphi _{n}/\delta _{n} ) \sharp ( g_{n}/\varepsilon _{n} ) ) =X_{\delta }^{ \beta } ( \varphi _{n}/\delta _{n} ) \star _{q}^{\tilde{q}}X_{\delta }^{\beta } ( g_{n}/\varepsilon _{n} )$.

Proof

(i) Let $\varphi _{n}/\delta _{n}\in H ( \mathbb{R} )$. Then $\varphi _{n}\sharp \delta _{m}=g_{m}\sharp \delta _{n}$ for all $m,n\in \mathbb{N}$. But then, by Theorem 3, we have $G_{\delta }^{\beta } (\varphi _{n}\sharp \delta _{m} )=G_{ \delta }^{\beta } (g_{m}\sharp \delta _{n} )$. Therefore, $G_{\delta }^{\beta }\varphi _{n}\star _{q}^{\tilde{q}}\delta _{m}=G_{\delta }^{\beta }g_{m}\star _{q}^{\tilde{q}}\delta _{n}$.

Hence, we have

$$ G_{\delta }^{\beta }\varphi _{n}/\delta _{n}\in B \bigl( \mathbb{R}^{2} \bigr) . $$

Now, we show that $X_{\delta }^{\beta }$ is independent of the representative. Let $\varphi _{n}/\delta _{n}=g_{n}/\varepsilon _{n}$ in $H ( \mathbb{R} )$, then $\varphi _{n}\sharp \varepsilon _{m}=g_{m}\sharp \delta _{n}$ for all $m,n\in \mathbb{N}$. Applying Theorem 3 gives $G_{\delta }^{\beta }\varphi _{n}\star _{q}^{\tilde{q}}\varepsilon _{m}=G_{ \delta }^{\beta }g_{m}\star _{q}^{\tilde{q}}\delta _{n}$. Hence, $G_{\delta }^{\beta }\varphi _{n}/\delta _{n}=G_{\delta }^{\beta }g_{n}/ \varepsilon _{n}$. Thus, $G_{\delta }^{\beta } ( \varphi _{n}/\delta _{n} ) =G_{ \delta }^{\beta } ( g_{n}/\varepsilon _{n} )$.

Proof (ii) Linearity of $X_{\delta }^{\beta }$ follows from linearity of $G_{\delta }^{\beta }$. Let $\varphi _{n}/\delta _{n},g_{n}/\varepsilon _{n}\in H ( \mathbb{R} )$ be such that $X_{\delta }^{\beta } ( \varphi _{n}/\delta _{n} ) =X_{ \delta }^{\beta } ( g_{n}/\varepsilon _{n} ) \in B ( \mathbb{R}^{2} )$. Then by Eq. (14) we get $G_{\delta }^{\beta }\varphi _{n}/\delta _{n}=G_{\delta }^{\beta }g_{n}/ \varepsilon _{n}$. This means that $G_{\delta }^{\beta }\varphi _{n}\star _{q}^{\tilde{q}}\varepsilon _{m}=G_{ \delta }^{\beta }g_{m}\star _{q}^{\tilde{q}}\delta _{n}$ for all $m,n\in \mathbb{N}$. Hence Theorem 3 gives $G_{\delta }^{\beta } ( \varphi _{n}\sharp \varepsilon _{m} ) =G_{\delta }^{\beta } ( g_{m}\sharp \delta _{n} )$. As $G_{\delta }^{\beta }:L^{1} ( \mathbb{R} ) \rightarrow L^{1} ( \mathbb{R}^{2} )$ is one-to-one, we have $\varphi _{n}\sharp \varepsilon _{m}=g_{m}\sharp \delta _{n}$ for all $m,n\in \mathbb{N}$. This in turn yields $\varphi _{n}/\delta _{n}=g_{n}/\varepsilon _{n}\in H ( \mathbb{R} )$. The onto condition is clear.

We prove Part (iv) as similar proofs for Part (iii) and the convolution theorem in Part (v) may be followed in [9, 12]. Let $\rho \in L_{loc}^{1} ( \mathbb{R}^{2} )$, then $( \rho \sharp \delta _{n} ) /\delta _{n}$ is the representative of ρ in $H ( \mathbb{R} )$, $\{ \delta _{n} \} \in \Delta$$( \forall n\in {\mathbb{N}} )$. Clearly, for all $n\in {\mathbb{N}}$, $\{ \delta _{n} \}$ is independent of the representative. Hence, by the convolution theorem, we get

$$ X_{\delta }^{\beta } \bigl( ( \rho \sharp \delta _{n} ) / \delta _{n} \bigr) =G_{\delta }^{\beta } ( \rho \sharp \delta _{n} ) /\delta _{n}= \bigl( G_{\delta }^{\beta } \rho \star _{q}^{ \tilde{q}}\delta _{n} \bigr) /\delta _{n}=G_{\delta }^{\beta }\rho \star _{q}^{\tilde{q}} ( \delta _{n}/\delta _{n} ) . $$

Thus, $( G_{\delta }^{\beta }\rho \star _{q}^{\tilde{q}}\delta _{n} ) /\delta _{n}$ is the representative of $G_{\delta }^{\beta }\rho$ in the space $L_{loc}^{1} ( \mathbb{R}^{2} )$.

The proof is therefore finished. □

We introduce the inverse operator of $X_{\delta }^{\beta }$ as follows.

Definition 13

Let $G_{\delta }^{\beta }\varphi _{n}/\delta _{n}\in B ( \mathbb{R}^{2} )$. We define the inverse $X_{\delta }^{\beta }$ integral of a Boehmian $G_{\delta }^{\beta }\varphi _{n}/\delta _{n}$ in $B ( \mathbb{R}^{2} )$ as follows:

$$ S_{\delta }^{\beta } \bigl( G_{\delta }^{\beta }\varphi _{n}/\delta _{n} \bigr) =\varphi _{n}/\delta _{n} $$

for each $\{ \delta _{n} \} \in \Delta$.

Theorem 14

Let $G_{\delta }^{\beta }\varphi _{n}/\delta _{n}\in B ( \mathbb{R}^{2} )$and $\varphi \in L_{loc}^{1} ( \mathbb{R}^{2} )$be given. We have

$$ S_{\delta }^{\beta } \bigl( \bigl( G_{\delta }^{\beta } \varphi _{n}/ \delta _{n} \bigr) \star _{q}^{\tilde{q}} \varphi \bigr) = ( \varphi _{n}/\delta _{n} ) \sharp \varphi \quad \textit{and}\quad X_{ \delta }^{\beta } \bigl( ( \varphi _{n}/ \delta _{n} ) \sharp \varphi \bigr) = \bigl( G_{\delta }^{\beta } \varphi _{n}/ \delta _{n} \bigr) \star _{q}^{\tilde{q}} \varphi . $$

Proof

Assume $G_{\delta }^{\beta }\varphi _{n}/\delta _{n}\in B ( \mathbb{R}^{2} )$. For every $\varphi \in L_{loc}^{1} ( \mathbb{R}^{2} )$, by using the convolution theorem and Definition 11, we have

$$\begin{aligned} S_{\delta }^{\beta } \bigl( \bigl( G_{\delta }^{\beta } \varphi _{n}/ \delta _{n} \bigr) \star _{q}^{\tilde{q}} \varphi \bigr) =&S_{ \delta }^{\beta } \bigl( \bigl( G_{\delta }^{\beta }\varphi _{n} \star _{q}^{\tilde{q}} \varphi \bigr) /\delta _{n} \bigr) \\ =&S_{\delta }^{\beta } \bigl( G_{\delta }^{\beta } ( \varphi _{n} \sharp \varphi ) \bigr) /\delta _{n} \\ =& ( \varphi _{n}\sharp \varphi ) /\delta _{n} \\ =& ( \varphi _{n}/\delta _{n} )\sharp \varphi . \end{aligned}$$

As the proof of the second part is similar, we omit the details. This completely finishes the proof of the theorem. □