On a Durrmeyer-type modification of the Exponential sampling series

In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described.

enables one to reconstruct functions (signals) which are Mellin-bandlimited. Indeed, the study of the structure of the class of Mellin-bandlimited functions is a deep topic of Mellin analysis. It was studied in [5,6], in terms of Mellin-Bernstein spaces.
As for the Shannon sampling series, later on a generalized version of the exponential sampling series was introduced in [9] (see also [3,15]), in which the "sinc-log" kernel is replaced by a function defined on ℝ + satisfying suitable assumptions. This is very important in order to obtain reconstructions of functions not necessarily Mellin band-limited, and to develop a "prediction" theory as a counterpart of the theory developed in [8] for the generalized sampling series of Fourier analysis.
In [7] the classical generalized sampling series of Fourier analysis was modified by replacing the sample values of the function f with its mean-value in small intervals, so defining the Kantorovich sampling series. This represents a wide field of investigations, due to its practical applications in various sectors of applied sciences (see e.g. [1,2,[20][21][22][23][24][25] and references therein).
In [13] (see also [10] and [14] for a multivariate version), a further extension of the generalized Kantorovich sampling series was introduced, by replacing the mean-values of the function f by "approximating values" of f defined through general convolution operator, obtaining the so-called Durrmeyer generalized sampling type series. Note that a general approach to sampling series in functional spaces was developed in [31].
In the present paper we present an analogous generalization for the exponential sampling series, using Mellin convolution operators. The present paper represents a first step in the construction of the approximation theory for semi-discrete operators in Mellin setting: here we obtain pointwise and uniform convergence theorems, giving also a quantitative version in terms of the so-called log-modulus of continuity (see Sect. 5), and an asymptotic formula of Voronovskaja type under certain local regularity assumptions on the function f. The last section is devoted to some examples, involving the central B-splines (see [8,15,30]) and the Mellin-Jackson kernel (see [12]). Many other examples can be constructed.

Preliminaries
Let us denote by ℕ , ℕ 0 and ℤ the sets of positive integers, nonnegative integers and integers respectively. Moreover by ℝ and ℝ + we denote the sets of all real and positive real numbers respectively.
In what follows, for simplicity, we will assume that the functions f defined on ℝ + take their values in ℝ , but the results remain valid also for complex-valued functions.
Let L ∞ (ℝ + ) be the space of all the essentially bounded functions f ∶ ℝ + → ℝ endowed with the usual norm ‖f ‖ ∞ . Moreover we will denote by C = C(ℝ + ) the space of all the continuous functions f ∶ ℝ + → ℝ and by C 0 = C 0 (ℝ + ) the space of all the uniformly continuous and bounded functions on ℝ + .
We say that a function f ∈ C(ℝ + ) is "log-uniformly continuous" on ℝ + , if for any > 0 there exists > 0 such that We denote by C(ℝ + ) the space containing the log-uniformly continuous and bounded functions f ∶ ℝ + → ℝ. Note that in general a log-uniformly continuous function is not necessarily uniformly continuous and conversely. Obviously the two notions are equivalent on compact intervals in ℝ + .
For a function f ∈ C 0 and r ∈ ℕ we will say that f belongs to C r locally at a point x ∈ ℝ + if there is a neighbourhood U of x such that f is (r − 1)-fold continuously differentiable in U and f (r) (x) exists.
The pointwise Mellin differential operator , or the pointwise Mellin derivative f of a function f ∶ ℝ + → ℝ, is defined by (see [17]) provided f � (x) exists on ℝ + . The Mellin differential operator of order r ∈ ℕ is defined iteratively by For convenience set 0 ∶= I, I denoting the identity. For instance, the first three Mellin derivatives are given by: In general, we have (see [17]) where S(r, k), 0 ≤ k ≤ r, denote the (classical) Stirling numbers of second kind. We have the following Taylor formula with Mellin derivatives (see [11,28]).
Proposition 1 For any f ∈ C 0 (ℝ + ) belonging to C r locally at a point x ∈ ℝ + we have where h ∶ ℝ + → ℝ is a bounded function such that h(t) → 0 for t → 1.

Remark 1 The boundedness of the function h in the remainder of the Taylor formula with
Mellin derivatives comes from the boundedness of the function f. However, the same holds for functions f ∈ C(ℝ + ) which have a growth of type for positive constants a, b and x ∈ ℝ + . Indeed, employing the limit relation at the point 1 the function h is obviously bounded in an interval containing 1, while using the growth condition on f and expressing h in terms of the Taylor formula, one can see easily the boundedness of h in the complement of the interval.

Exponential sampling Durrmeyer operator
Let ∶ ℝ + → ℝ be a continuous function such that the following assumptions are satisfied uniformly with respect to u ∈ ℝ + ; we denote by the class of all functions satisfying the above assumptions. Let ∶ ℝ + → ℝ be a function with the following conditions and let us denote by Ψ the class of all functions satisfying the above assumptions. Let ∈ ℕ 0 . For x ∈ ℝ + we define the algebraic moments of order of ∈ and ∈ Ψ as and The absolute moments of order of ∈ and ∈ Ψ are defined as The same for the absolute moments of ∈ Ψ. Indeed we may write for < Let ∈ and ∈ Ψ. For any n ∈ ℕ, and f ∶ ℝ + → ℝ , we define the exponential sampling Durrmeyer series as for x ∈ ℝ + and for any function f ∈ dom S , n , being dom S , n the set of all functions f for which the series is absolutely convergent at every x. Using the conditions of the classes and Ψ, it is easy to see that the above operator is well defined as an absolutely convergent series, for any function f ∈ L ∞ (ℝ + ). In particular C 0 (ℝ + ) ⊂ dom S , n , for any n ∈ ℕ. Indeed, putting t = e −k u n We can determine larger subspaces of the domain of S , n . We admit functions which grow like a power of the logarithm. We have the following (see also [15]) Proof We prove the proposition considering r = 2 because the general case follows in an Then, using the change of variable (te k ) 1∕n = u in the above integral, we obtain Taking in to account that k 2 ≤ 2((k − log x n ) 2 + log 2 x n ) and |k| ≤ |k − log x n | + | log x n | we obtain The general case is obtained by (| log t| + |k|) r ≤ 2 r−1 (| log t| r + |k| r ).

Pointwise and uniform convergence
In this section we state a pointwise convergence theorem at continuity points of the function f. Then we obtain as a corollary, a uniform convergence theorem for functions belonging to C(ℝ + ).
We begin with the following pointwise convergence theorem.
Theorem 1 Let ∈ Ψ and ∈ . Let f be a bounded function. If x ∈ ℝ + is a continuity point for f then Proof Let x ∈ ℝ + be a continuity point of f. Since ∈ and ∈ Ψ , we have As to I 2 we have, by the boundedness of f We have For I 1,1 2 since | log(e −k x n )| < n ∕2 we have x n e −n e −k < e −n e n ∕2 = e −n ∕2 and so for fixed x, we obtain and for the absolute continuity of the integral of we have that, for large n x n e n e −k � (t)� dt t = I 1 2 + I 2 2 .
for every k such that | log(e −k x n )| < n ∕2. As to I 1,2 2 and so for large n using property ( .3 ) we have I 1,2 2 ≤ 2‖f ‖ ∞M0 ( ) . The term I 2 2 is is estimated in a similar way, so the assertion follows.
Using essentially the same reasoning employed in the previous theorem we can prove the following uniform convergence result.

An asymptotic formula
In this section we will assume that ∈ and ∈ Ψ are such that their moments m ( , u) and m ( ) up to the order r ∈ ℕ are finite and moreover m ( , u) = m ( ) are all independent of u ∈ ℝ + , for = 1, 2, … r.
We have the following result Theorem 3 Let ∈ , ∈ Ψ be such that M r ( ) and M r ( ) are both finite. Let f ∈ L ∞ (ℝ + ) and let x ∈ ℝ + be fixed. If for r ∈ ℕ, f ∈ C r locally at the point x Proof Using Proposition 1 we have where h is a bounded function and h(y) → 0 for y → 1. Thus For every j = 1, 2, … , r we have, with the change of variable e −k u n = t, Now we evaluate the term Let > 0 be fixed and let be such that |h(t)| < whenever | log t| < . So we have For R 1 we have, by the same change of variable as before, We consider only the term R 1 2 , since the other terms can be estimated in a similar way. We have For R 1,1 2 since | log(e −k x n )| < n ∕2 in an analogous way to Theorem 1 we can obtain and so Note that satisfies all the classical properties of a modulus of continuity. In particular we will employ the following one: for every , > 0.
We have the following quantitative estimate The values of the algebraic moments can be deduced by the Mellin-Poisson summation formula, and we have (see [9]) that only for j = 0, 1 the corresponding algebraic moments are independent of x. Moreover In this case the asymptotic formula reduces to 2) Let us consider the generalized Mellin-Jackson kernel, which is defined by (see [12]) where x ∈ ℝ + , ∈ ℕ, ≥ 1, d , is a normalization constant, i.e.  It is known that [J , ] ∧ M (iv) = 0 for |v| ≥ 1∕ , thus J , is Mellin band-limited. We put = J , (x). Using the Mellin-Poisson summation formula (see [9]) we obtain So we can prove that assumptions ( .1 ), ( .2 ) and ( .3 ) are satisfied, see [9]. Concerning the moments, using again the Mellin-Poisson summation formula for the derivatives as in [9], one has m 1 (J , ) = 0, and, for > 3∕2, (see [12]) Moreover, M 2 (J , ) < +∞. For the function we put (x) = B 2 (x) the previous spline. We have that Therefore the assumptions of the previous theorems are satisfied, with r = 2. In particular we obtain the following Voronovskaja formula, for f ∈ C(ℝ + ) of class C (2) locally at the point x ∈ ℝ + ∶