Semi‐discrete operators in multivariate setting: Convergence properties and applications

In this paper, we study the convergence properties of certain semi‐discrete exponential‐type sampling series in a multidimensional frame. In particular, we obtain an asymptotic formula of Voronovskaya type, which gives a precise order of approximation in the space of continuous functions, and we give some particular example illustrating the theory. Applications to the study of the seismic waves are illustrated.


INTRODUCTION
A multivariate version of the classical generalized sampling series, in the realm of Fourier analysis, was introduced in Butzer et al. [1] and next developed in Bardaro et al. [2] in view of their concrete applications in image reconstruction for which it represents an appropriate mathematical model. It is defined as where x ∈ R N , w = (w 1 , … , w N ), with w > 0, for every = 1, … , N, is a suitable kernel function and belongs to a suitable functional space. As in the univariate case, the computation of the sampled values (k∕w) depend on experimental measurements that contain errors of approximation. To avoid this problem, some generalized versions of the sampling series (1) were introduced, in which the sampled values are replaced by a "mean" defined through a convolution between and another kernel function satisfying classical assumptions of the approximate identities. This leads to sampling operators of type (see Bardaro and Mantellini [3]): Starting from the 1980s of the last century, with the aim to obtain mathematical models for the study of problems in optical physics, like, for example, light-scattering and diffraction, certain sampling series, in which the samples are not equally spaced over the real line, but now exponentially spaced over the positive real line, in such a way that they accumulates at zero, were introduced in Bertero and Pike [8] and Parker et al. [9], and later on deeply studied in a rigorous way [10]. They are called (univariate) "exponential sampling series." The generalized version of such series was introduced in Bardaro et al. [11]. Later on Kantorovich [12] versions were studied in previous studies [13][14][15] while a Durrmeyer version was introduced in Bardaro and Mantellini [16] (see also Bajpeyi et al. [17]). In Bardaro et al. [18], a two-dimensional version was studied with the aim to obtain mathematical models for the study of the propagation of seismic waves (a general multivariate version was recently studied in Kursun et al. [19]). However, this kind of applications is mainly meaningful in three-dimensional case. Therefore, in the present paper, we discuss a multivariate version of the generalized exponential sampling series of Durrmeyer type, defined through a Mellin-type convolution, as We establish pointwise and uniform convergence theorems and an asymptotic formula which gives an exact order of pointwise approximation. The proof of this last formula is based on a multivariate Mellin-Taylor formula here established in Section 4. Note that the particular case of Kantorovich exponential sampling series in univariate case was studied in Angamuthu and Bajpeyi and Aral et al. [13,14]. Then, we also consider global estimates of the uniform convergence by means a modulus of continuity, which is suitable in Mellin analysis. Section 9 contains some examples of kernels to illustrate the theory developed. Finally, the last section contains a study of the propagation of seismic waves in three-dimensional case. The advantage in the use of this kind of sampling operators is that with the use of the convolution we can implement a filter: If we perturb the function (signal), through the filtering operation (convolution), we still get good approximations, which is not possible with the usual generalized exponential sampling series.

NOTATIONS AND BASIC DEFINITIONS
We denote by N N , N N 0 , and Z N the sets of N-tuples k = (k 1 , … , k N ) of positive integers, nonnegative integers, and integers, respectively, and by R N and R N + the sets of N-tuples of real and positive real numbers, respectively. We set []k[] ∶= k 1 + … + k N . For given vectors x = (x 1 , … , x N ), y = ( 1 , … , N ) ∈ R N , we define the relation x > y iff x i > i for every i = 1, … , N. We set 1 ∶= (1, … , 1) ∈ R N , 0 ∶= (0, … , 0) ∈ R N , and by e i , i = 1, … , N, we denote the vectors of the canonical basis of R N + . As to the operation with vectors, we write x + y ∶= (x 1 + 1 , … , x N + N ) and for ∈ R, Moreover, we will use the following notations: by ||x||, we denote the euclidean norm of x and d( a nonempty open set. By C(J), we denote the space comprising all the bounded and uniformly continuous functions over J and by C c (J) the subspace of C(J) containing the functions with compact support.
We denote by (J) the space comprising all the log-uniformly continuous and bounded functions over J. Let us remark that if J is compact, the two notions of uniform continuity and log-uniform continuity are equivalent. The two notions are different for non-compact sets.
We will say that a function ∶ J → C belongs to C (r) (J) locally at the point x ∈ J if there is a neighborhood I of x such that is (r − 1)-times differentiable on I and all the partial derivatives exist. Now, we introduce the partial derivatives in Mellin setting (see [20]). The first partial Mellin derivative of a function at the point where for a sake of simplicity, we set Θ

MELLIN-TAYLOR FORMULAE
In this section, we establish the Mellin-Taylor formulae, which represent the Mellin counterpart of the classical Taylor formulae in the setting of Mellin analysis, employing the Mellin derivatives. In order to do that, we introduce the following notation: given with m ∈ N and ∈ C (m) (R N + ) locally at the point (x 1 , … , x N ). Note that two-dimensional versions were given in Bardaro et al. [18]. For completeness, we give the proof which can be obtained following similar reasonings.
with the Lagrange remainder is a suitable point in the segment with endpoints x and tx.
Proof. We prove only the case m = 2. Let us define the function Applying the uni-dimensional Mellin-Taylor formula (see Bardaro and Mantellini [21]) with the Lagrange remainder, we get witht ∈]1, e[. By definition, we have ΘF(t) = tF ′ (t), thus and for t = 1, we have Analogously, for Θ 2 F(t) = tF ′ (t) + t 2 F ′′ (t), it holds , which belongs to the segment with endpoints x e tx, we get Now, using the definition of Mellin partial derivative (3), we have 2 Hence, Substituting t = e in (5), we obtain Therefore, The case m = 1 is treated in the same way.
Proof. We prove the proposition for m = 2. We have By (4), we obtain Hence, The assertion follows taking into account that Λ belong to the segment with endpoints x e tx and ∈ C (2) (R N + ). □ For m = 2, we can rewrite the local Mellin-Taylor formula as with the Peano remainder Note that this remainder can be written also in the form

DURRMEYER EXPONENTIAL SAMPLING SERIES
In this section, we introduce our semi-discrete exponential sampling operators, namely, the so-called multidimensional Durrmeyer-type exponential sampling operators. We premise some further definitions and notations.

Definition 1.
Let Φ (N) be the class of all functions , with ∶ R N + → R, satisfying the following conditions: The absolute moments of order j of ∈ Φ (N) are defined by Thus, assumption ( .2) becomes a particular case of ( .4).
We denote by Φ (N) the class of all functions such that ( .1), ( .3), and ( .4) holds. Next, we denote by Ψ (N) the class comprising all the measurable functions ∶ R N + → R, such that the following conditions hold: and the absolute moment of order j of ∈ Ψ (N) as Moreover, for ∈ Ψ (N) and ∈ Φ (N) , we set Remark 2. Note that for every ∈ N 0 , and ∈ Φ (N) , it holds Indeed, for ∈ N 0 and j = ( Moreover, by the inequality Analogously, for every ∈ N 0 , for the absolute moment of ∈ Ψ (N) , we havẽ Now, we introduce the multivariate Durrmeyer exponential sampling series. First, we denote by  the set comprising all measurable functions ∶ R N + → C such that the integrals are well defined as Lebesgue integrals for every k ∈ Z N and w = (w 1 , … , w N ) > 0.
, and a measurable ∶ R N + → C, we define the Durrmeyer exponential sampling series of as denotes the set of all functions ∈  for which the series is absolutely convergent for every x.
In particular, L ∞ (R N + ) ⊂ dom S , . Indeed, using the assumptions on the functions (N) and (N) and setting t = e −k u w , However, we can determine larger subspaces domS , . Indeed, we have the following: Proof. We give the proof for = 2. In this case, the growth condition on the function can be written in the form Thus, setting t = e −k u w , we get Since ∈ Φ (N) 2 , and ∈ Ψ (N) , using the inequalities (7) and (8), for I 1 , I 2 , and I 3 , we have As to I 2 , since we obtain The term I 3 can be estimated as Therefore, ) .

CONVERGENCE THEOREMS
We begin with the following pointwise and uniform convergence results for the family of operators S , w .
Proof. Let x ∈ R N + be a continuity point of . Making the substitution t = e −k u w in the integral, we have By continuity of at the point x ∈ R N + for a given > 0, there is = ( ) > 0 such that | (u) − (x)| < , whenever || log u − log x|| = || log( u x )|| < . Hence, for this , we can write As to I 1 , one has For I 2 , taking into account the boundedness of , we get We estimate the term I 2,1 . By ( .2) as w → ∞, we have As to I 2,2 , since for every k such that ||k − log x w || ≤ ||w|| 2 , one has ||w|| ≤ || log t + k − log x w || ≤ || log t|| + ||k − log x w || ≤ || log t|| + ||w|| 2 , we deduce that || log t|| ≥ ||w|| 2 . Moreover, by the absolute continuity of the Lebesgue integral for sufficiently large ||w||, we obtain and the assertion follows. □ Now, using similar reasonings of the previous proof, we can prove the following uniform convergence result.

AN ASYMPTOTIC FORMULA
In this section, we establish an asymptotic formula for our operator, which gives a precise order of pointwise approximation, under certain local regularity assumptions on the function . In order to do that, we premise some further notations.
For vectors j = ( 1 , … , N ) and h = (h 1 , … , h N ) ∈ N N , with j ≤ h, a ∈ R N , we set We have the following Theorem 3. Let ∈ Φ (N) r with r = 1, 2 and ∈ Ψ (N) be such that m j ( ),m j ( ) < +∞ for every ≤ r. If ∈ L ∞ (R N + ) belongs to C (r) locally at x ∈ R N + , then for every w → ∞, we have Proof. We give the proof for r = 2. Let x ∈ R N + be fixed, and let be of class C (2) locally at x ∈ R N + . Using the local Mellin-Taylor formula (6), with m = 2, a bounded function H exists such that lim t→1 H(t) = 0 and Thus, for the given functions ∈ Φ (N) 2 and ∈ Ψ (N) , we have || log u − log x|| 2 du ⟨u⟩ =∶ I 1 + I 2 + J.
As to the terms I with = 1, 2 setting t = e −k u w , one has Now, with the above notations, we get Thus, Let us estimate now the term J. For a fixed > 0, there is = ( ) > 0 such that As to J 1 , by (7) and (8), the assumptions ∈ Φ (N) 2 and ∈ Ψ (N) and the inequality (w>0) for sufficiently large w, we obtain As to J 2 , by the boundedness of H, we obtain the estimate Then, terms J 2,1 , J 2,2 , J 2,3 can be estimated using analogous reasonings as above. For example, let us consider J 2,1 . We have Now, for k such that ||k − log x w || ≤ ||w|| 2 , one has and for the absolute continuity of the Lebesgue integral, for every w with sufficiently large ||w||, one has and so Analogously, we obtain Summing up, we finally obtain where The proof is now complete. □ A consequence of Theorem 3 is the following Voronovskaja-type formula. We premise the following assumption. For a given vector w ∈ R N + : Setting a = (a 1 , … , a N ) under the above assumption (1), we have We have the following: please put a space between (x) and (||k|| = r) ) .
The assertion follows. □

A QUANTITATIVE ESTIMATE
We first introduce the logarithmic modulus of continuity of a function ∈ (R N + ) on setting for every > 0 The above modulus satisfies all the classical assumptions, in particular, it is a increasing function of > 0, and it holds for every > 0.
In this section, we obtain a quantitative estimate of the convergence of (S , w )(x) to (x) in terms of ( , ).
Theorem 4. Let ∈ Φ (N) and ∈ Ψ (N) be such that M 1 ( ) < ∞ andM 1 ( ) < ∞. If ∈ (R N + ), then for every > 0 and w>0, we have Proof. We have Now, for every > 0, using (13) with = || log (u/x)|| , we obtain For sufficiently large w, employing the change of variable t = e −k u w , we have thus, the assertion follows. □ As an immediate consequence, we get the following corollary.

Corollary 2.
Under the assumptions of Theorem 4, we have where A is a constant depending only on and .

SOME EXAMPLES
In this section, we discuss some examples of kernels and satisfying the assumptions employed in the previous sections. For details, see Bardaro et al. [18].

Mellin-splines
Denoting by r + the positive part of the number r ∈ R, for n ∈ N, we define the one-dimensional Mellin-spline of order n as (see Bardaro et al. [11]) These functions are the Mellin version of the classical central B-splines (see, e.g., Schumaker [22]). As an example, let us put (x) = (x) = B 2 (x) ∶= (1 − | log x|) + .
Since B 2 has a compact support, all the absolute moments M (B 2 ) are finite, while the values of the algebraic moments can be deduced employing the Mellin-Poisson summation formula, and we have (see Bardaro et al. [11]) that for = 0, 1, the corresponding moments are independent of x, and As to the function , we havem Now, we define ( Figure 1).
For the algebraic moments of B 2,2 , we have  We can apply Corollary 1 for (x 1 , ) .

SEISMIC WAVES AND EXPONENTIAL SAMPLING BY MEANS OF DURRMEYER OPERATORS
The application advantage introduced by the Durrmeyer class of operators stands in their capability to reconstruct and filter a signal starting from its samples, such to suppress the undesired noise whose presence is strictly connected with any measurement process. Moreover, it is possible to formalize different type of filters, if they can be expressed by the kernel function under the integral part of the operator. Suppressing noise is one of the most important problems faced in physical applications. Moreover, as previously stated in Bardaro et al. [18], exponential sampling is particular suitable for the description of the seismic waves. Thanks to the previous observations, the application of exponential sampling Durrmeyer operators appears a natural choice in the context of the mathematical modeling of the seismogram. In fact, for the measurement of the amplitude of a earthquake, different Wood-Anderson seismographs are located in specific registration laboratories. When an earthquake occurs, the seismographs activate, registering the amplitude of the seismic waves in the three direction of the space: This registration can be quantified according to the Richter scale [24,25]. A suitable geophysical model, used to mathematically describe the local magnitude of the seismic waves, by means of the data measured by the seismographs, is given by L M (R, ) = log 10 A M (R, ) + 1.79 log 10 R − 0.58, where A M (R, ) is the measured amplitude of the seismic wave, is a generic direction of propagation, and R is the distance of the seismograph from the epicenter, that is, from the point on the terrestrial surface that is firstly reached by the underground seismic waves (further details on mathematical modeling of seismic waves can be found in Bobbio et al. [26]). In case of invertibility of Equation (17) We used this expression in our simulations to estimate the quality of the results achieved by means of the Durrmeyer operators. In our reconstructions, the Durrmeyer operators have been equipped with the following discrete Mellin-spline  (1 + log(i))(1 + log( )), i e −1 < i ≤ 1, e −1 < ≤ 1, (1 + log(i))(1 − log( )), i e −1 < i ≤ 1, 1 ≤ < e, (1 − log(i))(1 + log( )), i 1 ≤ i < e, e −1 < ≤ 1, (1 − log(i))(1 − log( )), i 1 ≤ i < e, 1 ≤ < e, 0, otherwise, (19) implemented in Matlab©, with i, ∈ N being the coordinates of the discrete grid considered for the calculation. In Algorithm 1, the pseudocode is shown.
Moreover, the well-known mean square error (MSE) index has been used for numerical esteems. We recall that the MSE, in a two-dimensional discrete set, is defined as where I o , I r are two generic discrete functions assuming values, in the most general case, in C and defined on a grid of N × M points. Basically, the MSE is the square L 2 norm of I o − I r . In Figure 3, the original noise-free 2D seismic wave, corresponding to Equation (18) is shown. Note: As N and w increase, the quality of the reconstruction improves. To evaluate the filtering properties of the Durrmeyer operators, we have generated a random zero-mean Gaussian noise and, assuming an additive model, we have corrupted the original function, achieving the noisy samples we assume to be representative of the physical sampling process. In particular, the level of noise has been controlled by means of the signal-to-noise ratio (SNR), expressing the ratio between the useful signal and the undesired one. A value of SNR = 10 means the signal has an amplitude 10 time bigger than the noise, that is, the higher the SNR, the more measured signal is close to the noise-free function (see Figure 4).
In Table 1, the MSE for different combination of N and w and for different values of the SNR has been reported. The data show how, by increasing the values of N and w, the operator provides a better reconstruction.
Thanks to its filtering capability, the Durrmeyer operator reconstructions of the original signal, even in presence of considerable noise (SNR < 50), appears faithful to the original: the low values of the MSE, increasing the values of the parameters N and w is a quantitative proof of this behavior. In Figure 5, an example of reconstructions, both in 2D as in 1D, is provided.