1 Introduction

The theory of the exponential sampling series of (real or complex)-valued functions f defined over the positive real axis, is a powerful tool for investigating certain phenomena in optical physics, as for example, light scattering, Fraunhofer diffraction etc, (see e.g. [16, 19, 26, 29]). From a mathematical point of view these series were rigorously studied in [18], (see also [4]). The suitable frame for studying these operators is the Mellin analysis, in particular the Mellin transform theory (see [17, 28]). Indeed, the exponential sampling operator represents the counterpart of the classical Shannon sampling series of Fourier analysis (see [27]) in Mellin setting. Now the samples are not equally spaced, but exponentially spaced over the positive real axis, and the classical “sinc” kernel is now replaced by a composition of the sinc-function with the logarithm. The exponential sampling series 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 \(\varphi\) defined on \({\mathbb {R}}^+\) 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.

2 Preliminaries

Let us denote by \({\mathbb {N}}\), \({\mathbb {N}}_0\) and \({\mathbb {Z}}\) the sets of positive integers, nonnegative integers and integers respectively. Moreover by \({\mathbb {R}}\) and \({\mathbb {R}}^+\) 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 \({\mathbb {R}}^+\) take their values in \({\mathbb {R}}\), but the results remain valid also for complex-valued functions.

Let \(L^\infty ({\mathbb {R}}^+)\) be the space of all the essentially bounded functions \(f: {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\) endowed with the usual norm \(\Vert f\Vert _\infty .\)

Moreover we will denote by \(C=C({\mathbb {R}}^+)\) the space of all the continuous functions \(f: {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\) and by \(C^0 = C^0({\mathbb {R}}^+)\) the space of all the uniformly continuous and bounded functions on \({\mathbb {R}}^+.\)

We say that a function \(f\in C({\mathbb {R}}^+)\) is “log-uniformly continuous” on \({\mathbb {R}}^+,\) if for any \(\varepsilon >0\) there exists \(\delta >0\) such that

$$\begin{aligned} |f(u) - f(v)| < \varepsilon \,\,\text{ whenever }\, \,|\log u - \log v|\le \delta \quad (u, v\in {\mathbb {R}}^+). \end{aligned}$$

We denote by \(\mathcal {C}({\mathbb {R}}^+)\) the space containing the log-uniformly continuous and bounded functions \(f: {\mathbb {R}}^+ \rightarrow {\mathbb {R}}.\) 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 \({\mathbb {R}}^+.\)

For a function \(f\in C^0\) and \(r\in {\mathbb {N}}\) we will say that f belongs to \(C^r\) locally at a point \(x \in {\mathbb {R}}^+\) 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 \(\varTheta ,\) or the pointwise Mellin derivative \(\varTheta f\) of a function \(f: {\mathbb {R}}^+ \rightarrow {\mathbb {R}},\) is defined by (see [17])

$$\begin{aligned} \varTheta f(x) := x f'(x),~~x \in {\mathbb {R}}^+ \end{aligned}$$

provided \(f'(x)\) exists on \({\mathbb {R}}^+.\) The Mellin differential operator of order \(r \in {\mathbb {N}}\) is defined iteratively by

$$\begin{aligned} \varTheta ^1 := \varTheta ,\quad \quad \varTheta ^r := \varTheta (\varTheta ^{r-1}). \end{aligned}$$

For convenience set \(\varTheta ^0 := I,\) I denoting the identity. For instance, the first three Mellin derivatives are given by:

$$\begin{aligned} \varTheta f(x)&= xf'(x),\\ \varTheta ^2 f(x)&= x^2 f''(x) + xf'(x),\\ \varTheta ^3 f(x)&= x^3 f'''(x) + 3x^2f''(x) + xf'(x). \end{aligned}$$

In general, we have (see [17])

$$\begin{aligned} \varTheta ^r f(x) = \sum _{k=0}^r S (r,k) x^k f^{(k)}(x), \end{aligned}$$

where S(r, k),  \(0\le k\le r,\) denote the (classical) Stirling numbers of second kind. We have the following Taylor formula with Mellin derivatives (see [11, 28]).

Proposition 1

For any \(f\in C^0({\mathbb {R}}^+)\) belonging to \(C^r\) locally at a point \(x \in {\mathbb {R}}^+\) we have

$$\begin{aligned} f(tx) = f(x) + (\varTheta f)(x) \log t + \frac{(\varTheta ^2 f)(x)}{2!} \log ^2 t + \cdots + \frac{(\varTheta ^r f)(x)}{r!} \log ^r t + h(t) \log ^r t, \end{aligned}$$

where \(h:{\mathbb {R}}^+ \rightarrow {\mathbb {R}}\) is a bounded function such that \(h(t) \rightarrow 0\) for \(t\rightarrow 1.\)

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 \(f \in C({\mathbb {R}}^+)\) which have a growth of type

$$\begin{aligned} |f(x)|\le a + b |\log x|^r, \end{aligned}$$

for positive constants a, b and \(x \in {\mathbb {R}}^+.\) 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.

3 Exponential sampling Durrmeyer operator

Let \(\varphi :{\mathbb {R}}^+ \rightarrow {\mathbb {R}}\) be a continuous function such that the following assumptions are satisfied

(\(\varphi .1\)):

for every \(u \in {\mathbb {R}}^+,\) \(\sum \limits _{k= -\infty }^\infty \varphi (e^{-k}u) = 1;\)

(\(\varphi .2\)):

we have that

$$\begin{aligned} M_0 (\varphi )= \sup \limits _{u \in {\mathbb {R}}^+}\sum \limits _{k= -\infty }^\infty |\varphi (e^{ -k} u)|<+\infty ; \end{aligned}$$
(\(\varphi .3\)):

for some \(\nu \in {\mathbb {N}},\)

$$\begin{aligned} \lim \limits _{r \rightarrow +\infty } \sum \limits _{ |\log (e^{-k}x^n)|>r }|\varphi (e^{-k} u)| | k - \log u|^\nu =0. \end{aligned}$$

uniformly with respect to \(u\in {\mathbb {R}}^+;\)

we denote by \(\varPhi\) the class of all functions \(\varphi\) satisfying the above assumptions.

Let \(\psi :{\mathbb {R}}^+ \rightarrow {\mathbb {R}}\) be a function with the following conditions

(\(\psi .1\)):
$$\begin{aligned} \int _0^\infty \psi (t) \frac{dt}{t} =1; \end{aligned}$$
(\(\psi .2\)):
$$\begin{aligned} \widetilde{M}_0(\psi ) := \int _0^\infty |\psi (t) | \frac{dt}{t}< +\infty . \end{aligned}$$

and let us denote by \(\Psi\) the class of all functions \(\psi\) satisfying the above assumptions.

Let \(\nu \in {\mathbb {N}}_0.\) For \(x \in {\mathbb {R}}^+\) we define the algebraic moments of order \(\nu\) of \(\varphi \in \varPhi\) and \(\psi \in \Psi\) as

$$\begin{aligned}&m_\nu (\varphi , x): = \sum \limits _{k = -\infty }^\infty \varphi (e^{-k} x) \log ^\nu (e^{k}x^{-1}) = \sum \limits _{k = -\infty }^\infty \varphi (e^{-k} x)( k-\log (x) )^\nu \end{aligned}$$

and

$$\begin{aligned} \widetilde{m}_\nu (\psi ) := \int _0^\infty \psi (t) \log ^\nu t \, \frac{dt}{t}. \end{aligned}$$

The absolute moments of order \(\nu\) of \(\varphi \in \varPhi\) and \(\psi \in \Psi\) are defined as

$$\begin{aligned}&M_{\nu }(\varphi , x): = \sum \limits _{k = -\infty }^\infty |\varphi (e^{-k} x) | |\log ^\nu (e^{k}x^{-1}) |= \sum \limits _{k = -\infty }^\infty |\varphi (e^{-k} x)| |k -\log (x )|^\nu . \end{aligned}$$

We will set \(M_\nu (\varphi ):= \sup _{x \in {\mathbb {R}}^+} \sum \limits _{k = -\infty }^\infty |\varphi (e^{-k} x) | |\log ^\nu (e^{k}x^{-1}) |\) and

$$\begin{aligned} \widetilde{M}_\nu (\psi ) := \int _0^\infty |\log t|^\nu |\psi (t)| \frac{dt}{t}. \end{aligned}$$

Remark 2

Note that, for \(\varphi \in \varPhi ,\) if \(\mu , \nu \in {\mathbb {N}}\) with \(\mu < \nu ,\) then \(M_\nu (\varphi ) < \infty\) implies \(M_\mu (\varphi ) < \infty .\) Indeed for \(\mu < \nu\) we have

$$\begin{aligned} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x) | | k - \log x|^\mu&= \bigg ( \sum _{|k - \log x| < 1} + \sum _{| k - \log x| \ge 1} \bigg ) |\varphi (e^{-k} x) | | k - \log x|^\mu \\&\le M_0(\varphi ) + \sum _{| k - \log x| \ge 1} |\varphi (e^{-k} x) | | k - \log x|^\nu \\&\le M_0(\varphi ) + M_\nu (\varphi ). \end{aligned}$$

The same for the absolute moments of \(\psi \in \Psi .\) Indeed we may write for \(\mu < \nu\)

$$\begin{aligned} \int _0^\infty |\psi (t)| | \log t|^\mu \frac{dt}{t}&= \int _{|\log t| <1} |\psi (t)| | \log t|^\mu \frac{dt}{t} + \int _{|\log t| \ge 1} |\psi (t)| | \log t|^\mu \frac{dt}{t}\\&\le \widetilde{M}_0(\psi ) + \int _{|\log t| \ge 1} |\psi (t)| | \log t|^\nu \frac{dt}{t}\\&\le \widetilde{M}_0(\psi ) + \widetilde{M}_\nu (\psi ). \end{aligned}$$

Let \(\varphi \in \varPhi\) and \(\psi \in \Psi .\) For any \(n \in {\mathbb {N}},\) and \(f :{\mathbb {R}}^+ \rightarrow {\mathbb {R}}\), we define the exponential sampling Durrmeyer series as

$$\begin{aligned} (S^{\varphi , \psi }_{n} f)(x): = \sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) n \int _0^\infty \psi (e^{-k} u^n) f(u) \frac{du}{u} \end{aligned}$$
(1)

for \(x \in {\mathbb {R}}^+\) and for any function \(f \in \text{ dom }\,S^{\varphi , \psi }_{n},\) being \(\text{ dom }\,S^{\varphi , \psi }_{n}\) the set of all functions f for which the series is absolutely convergent at every x. Using the conditions of the classes \(\varPhi\) and \(\Psi ,\) it is easy to see that the above operator is well defined as an absolutely convergent series, for any function \(f\in L^\infty ({\mathbb {R}}^+).\) In particular \(C^0({\mathbb {R}}^+) \subset \text{ dom }\,S^{\varphi , \psi }_{n},\) for any \(n \in {\mathbb {N}}.\) Indeed, putting \(t= e^{-k} u^n\)

$$\begin{aligned} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| n \int _0^\infty |\psi (e^{-k} u^n)| |f(u)| \frac{du}{u} \le \Vert f\Vert _\infty \widetilde{M}_0(\psi ) M_0(\varphi ). \end{aligned}$$

We can determine larger subspaces of the domain of \(S^{\varphi , \psi }_{n}.\) We admit functions which grow like a power of the logarithm. We have the following (see also [15])

Proposition 2

Let \(\varphi \in \varPhi\) and \(\psi \in \Psi .\) Suppose that \(M_r(\varphi ) < \infty\) and \(\widetilde{M}_r(\psi ) < \infty\) for some \(r\in {\mathbb {N}}.\) If \(f : {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\) and \(|f(x)| \le a + b |\log x|^r\) for some \(a,b \in {\mathbb {R}}^+\) and all \(x\in {\mathbb {R}}^+,\) then f belongs to the domain of \(S^{\varphi , \psi }_{n}.\)

Proof

We prove the proposition considering r = 2 because the general case follows in an analogous way. So assume that \(|f(x)| \le a + b|\log x|^2\) with \(a, b \in {\mathbb {R}}^+.\)

$$\begin{aligned}&|(S^{\varphi , \psi }_{n} f)(x)| \le \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| n a \int _0^\infty |\psi (e^{-k} u^n)| \frac{du}{u}\\ & {\quad} + \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| n b \int _0^\infty |\psi (e^{-k} u^n)| |\log u|^2 \frac{du}{u}. \end{aligned}$$

Then, using the change of variable \((t e^k)^{1/n}= u\) in the above integral, we obtain

$$\begin{aligned}&|(S^{\varphi , \psi }_{n} f)(x)| \le a \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _0^\infty |\psi (t) |\frac{dt}{t} \\ & {\qquad}+ b \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _0^\infty |\psi (t) | |\log (t e^k)^{1/n}|^2 \frac{dt}{t}\\ & {\quad} \le a M_0(\varphi ) \widetilde{M}_0(\psi ) + \frac{b}{n^2} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _0^\infty |\psi (t) | (|\log t| + |k|)^2 \frac{dt}{t}\\ & {\quad}\le a M_0(\varphi ) \widetilde{M}_0(\psi ) + \frac{b}{n^2} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _0^\infty |\psi (t) | ( |\log t|^2 + 2 |k| |\log t| + k^2) \frac{dt}{t}\\ & {\quad}\le a M_0(\varphi ) \widetilde{M}_0(\psi ) + \frac{b}{n^2} M_0(\varphi ) \widetilde{M}_2 (\psi ) + \frac{2b}{n^2} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| |k| \int _0^\infty |\psi (t) | |\log t| \frac{dt}{t}\\ & {\qquad} +\frac{b}{n^2} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| k^2 \int _0^\infty |\psi (t) | \frac{dt}{t}\\ & {\quad} \le a M_0(\varphi ) \widetilde{M}_0(\psi ) + \frac{b}{n^2} M_0(\varphi ) \widetilde{M}_2 (\psi ) + \frac{2b}{n^2} \widetilde{M}_1 (\psi ) \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| |k|\\ & {\qquad} + \frac{b}{n^2} \widetilde{M}_0(\psi ) \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| k^2. \end{aligned}$$

Taking in to account that \(k^2 \le 2 ( (k- \log x^n)^2 + \log ^2 x^n)\) and \(|k| \le | k - \log x^n| + | \log x^n|\) we obtain

$$\begin{aligned}&|(S^{\varphi , \psi }_{n} f)(x)| \le a M_0(\varphi ) \widetilde{M}_0(\psi ) + \frac{b}{n^2} M_0(\varphi ) \widetilde{M}_2 (\psi ) + \frac{2b}{n^2} \widetilde{M}_1 (\psi ) M_1(\varphi ) \\ & {\quad} + \frac{2b}{n^2} \widetilde{M}_1 (\psi ) M_0(\varphi ) |\log x^n |+ \frac{b}{n^2} \widetilde{M}_0(\psi ) M_2(\varphi ) + \frac{b}{n^2} \widetilde{M}_0(\psi ) M_0(\varphi ) \log ^2 x^n. \end{aligned}$$

The general case is obtained by \(( |\log t| + |k|)^r \le 2^{r-1} ( |\log t|^r + |k|^r).\)

4 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 \(\mathcal {C}({\mathbb {R}}^+).\)

We begin with the following pointwise convergence theorem.

Theorem 1

Let \(\psi \in \Psi\) and \(\varphi \in \varPhi .\) Let f be a bounded function. If \(x\in {\mathbb {R}}^+\) is a continuity point for f then

$$\begin{aligned} \lim \limits _{n \rightarrow \infty } (S^{\varphi , \psi }_{n} f)(x) = f(x), \quad for\quad x \in {\mathbb {R}}^+. \end{aligned}$$
(2)

Proof

Let \(x\in {\mathbb {R}}^+\) be a continuity point of f. Since \(\varphi \in \varPhi\) and \(\psi \in \Psi\), we have

$$\begin{aligned} |(S^{\varphi , \psi }_{n} f)(x) - f(x)| \le \sum _{k = -\infty }^\infty | \varphi (e^{-k} x^n)| n \int _0^\infty |\psi (e^{-k} u^n)| |f(u) - f(x)| \frac{du}{u} \end{aligned}$$

For a fixed \(\varepsilon >0\) by the continuity of f at x, there exists \(\delta =\delta (\varepsilon )>0\) such that if \(|\log u - \log x| = |\log \frac{u}{x} |<\delta\), then \(|f(u) - f( x) |<\varepsilon .\) We write

$$\begin{aligned}&|(S^{\varphi , \psi }_{n} f)(x) - f(x)| \\ & {\quad}\le \sum _{k = -\infty }^\infty | \varphi (e^{-k} x^n)| n \bigg \{ \int _{ |\log (\frac{u}{x})|<\delta } + \int _{ |\log (\frac{u}{x})|\ge \delta } \bigg \} | \psi (e^{-k} u^n)| |f(u) - f(x)| \frac{du}{u}\\ & {\quad}:= I_1+I_2. \end{aligned}$$

Setting \(e^{-k} u^n=t\) by assumptions (\(\varphi .2\)) and (\(\psi .2\))

$$\begin{aligned}&I_1 = \sum _{k = -\infty }^\infty | \varphi (e^{-k} x^n)| n \int _{ |\log (\frac{u}{x})|<\delta } |\psi (e^{-k} u^n)| |f(u) - f(x)| \frac{du}{u} \\ & {\quad}\le \varepsilon \sum _{k = -\infty }^\infty | \varphi (e^{-k} x^n)| \int _{| \log (\frac{(t e^k)^{\frac{1}{n}}}{x})| <\delta } |\psi (t)| \frac{dt}{t} \le \varepsilon M_0(\varphi ) \widetilde{M}_0(\psi ). \end{aligned}$$

As to \(I_2\) we have, by the boundedness of f

$$\begin{aligned} I_2\le & {} \sum _{k = -\infty }^\infty | \varphi (e^{-k} x^n)| n \int _{ |\log (\frac{u}{x})|\ge \delta } |\psi (e^{-k} u^n)| |f(u) - f(x)| \frac{du}{u} \\ & {\quad} \le 2 \Vert f\Vert _\infty \sum _{k = -\infty }^\infty | \varphi (e^{-k} x^n)| \int _0^{x^n e^{-n\delta } e^{-k}} |\psi (t)| \frac{dt}{t}\\ & {\qquad} + 2 \Vert f\Vert _\infty \sum _{k = -\infty }^\infty | \varphi (e^{-k} x^n)| \int _{x^n e^{n\delta } e^{-k}}^\infty |\psi (t)| \frac{dt}{t}\\ & {\quad}= I^1_2 + I^2_2. \end{aligned}$$

We have

$$\begin{aligned} I^1_2&= 2 \Vert f\Vert _\infty \sum _{k = -\infty }^\infty | \varphi (e^{-k} x^n)| \int _0^{x^n e^{-n\delta } e^{-k}} |\psi (t)| \frac{dt}{t}\\ & {\quad}= 2 \Vert f\Vert _\infty \bigg \{ \sum _{ |\log (e^{-k}x^n)|< n\delta /2} + \sum _{ |\log (e^{-k}x^n)| \ge n \delta /2} \bigg \} | \varphi (e^{-k} x^n)| \int _0^{x^n e^{-n\delta } e^{-k}} |\psi (t)| \frac{dt}{t}\\ & {\quad}:= I_2^{1,1}+I_2^{1,2}. \end{aligned}$$

For \(I^{1,1}_2\) since \(|\log (e^{-k}x^n)|< n\delta /2\) we have \(x^n e^{-n\delta } e^{-k} < e^{-n \delta } e^{n \delta /2}= e^{-n\delta /2}\) and so for fixed \(x, \delta\) we obtain

$$\begin{aligned} \int _0^{x^n e^{-n\delta } e^{-k}} |\psi (t)| \frac{dt}{t} \le \int _0^{e^{-n\delta /2}} |\psi (t)| \frac{dt}{t} \end{aligned}$$

and for the absolute continuity of the integral of \(\psi\) we have that, for large n

$$\begin{aligned} \int _0^{x^n e^{-n\delta } e^{-k}} |\psi (t)| \frac{dt}{t} \le \varepsilon . \end{aligned}$$

for every k such that \(|\log (e^{-k}x^n)|< n\delta /2.\) As to \(I^{1,2}_2\)

$$\begin{aligned} I_2^{1,2}&= 2 \Vert f\Vert _\infty \sum _{ |\log (e^{-k}x^n)|\ge n\delta /2} | \varphi (e^{-k} x^n)| \int _0^{x^n e^{-n\delta } e^{-k}} |\psi (t)| \frac{dt}{t} \\ & < 2\Vert f\Vert _{\infty }\widetilde{M}_0(\psi ) \sum _{ |\log (e^{-k}x^n)|\ge n\delta /2} | \varphi (e^{-k} x^n)| \end{aligned}$$

and so for large n using property (\(\varphi .3\)) we have \(I_2^{1,2} \le 2\Vert f\Vert _{\infty } \widetilde{M}_0(\psi ) \varepsilon\). 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.

Theorem 2

Let \(f \in \mathcal {C}({\mathbb {R}}^+)\) \(\psi \in \Psi\) and \(\varphi \in \varPhi\) then

$$\begin{aligned} \lim \limits _{ n \rightarrow \infty } \big \Vert (S^{\varphi , \psi }_{n} f)(x) - f(x) \big \Vert _{\infty }=0. \end{aligned}$$
(3)

5 An asymptotic formula

In this section we will assume that \(\varphi \in \varPhi\) and \(\psi \in \Psi\) are such that their moments \(m_\nu (\varphi ,u)\) and \(\widetilde{m}_\nu (\psi )\) up to the order \(r \in {\mathbb {N}}\) are finite and moreover \(m_\nu (\varphi , u) = m _\nu (\varphi )\) are all independent of \(u \in {\mathbb {R}}^+,\) for \(\nu =1,2,\ldots r.\)

We have the following result

Theorem 3

Let \(\varphi \in \varPhi , \psi \in \Psi\) be such that \(M_r(\varphi )\) and \(\widetilde{M}_r(\psi )\) are both finite. Let \(f\in L^\infty ({\mathbb {R}}^+)\) and let \(x\in {\mathbb {R}}^+\) be fixed. If for \(r\in {\mathbb {N}},\) \(f\in C^r\) locally at the point x

$$(S^{\varphi , \psi }_n f) (x) = \sum _{j= 0}^r \frac{(\varTheta ^j f)(x)}{j! n^j} \sum _{\nu =0}^j \left( \begin{array}{c} j \\ \nu \end{array} \right) m_{j- \nu }(\varphi ) \widetilde{m}_\nu (\psi ) + o(n^{-r}), \quad n\rightarrow +\infty .$$

Proof

Using Proposition 1 we have

$$\begin{aligned} f(u) = f(x) + (\varTheta f)(x) \log \big (\frac{u}{x}\big ) + \frac{(\varTheta ^2 f)(x)}{2!} \log ^2 \big (\frac{u}{x}\big ) + \cdots + \frac{(\varTheta ^r f)(x)}{r!} \log ^r \big (\frac{u}{x}\big ) + h(\frac{u}{x}) \log ^r\big ( \frac{u}{x}\big ), \end{aligned}$$

where h is a bounded function and \(h(y) \rightarrow 0\) for \(y \rightarrow 1.\) Thus

$$\begin{aligned} (S^{\varphi , \psi }_{n} f)(x) - f(x)= & {} \sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) n \int _0^\infty \psi (e^{-k} u^n) (f(u) - f(x) ) \frac{du}{u} \\= & {} \sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) n \int _0^\infty \psi (e^{-k} u^n) (\varTheta f)(x) \log \big (\frac{u}{x}\big ) \frac{du}{u}\\ & {\quad} +\sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) n \int _0^\infty \psi (e^{-k} u^n) \frac{(\varTheta ^2 f)(x)}{2!} \log ^2\big (\frac{u}{x}\big ) \frac{du}{u}\\& + \cdots +\sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) n \int _0^\infty \psi (e^{-k} u^n) \frac{(\varTheta ^r f)(x)}{r!} \log ^r\big (\frac{u}{x}\big )\frac{du}{u}\\& {\quad} + \sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) n \int _0^\infty \psi (e^{-k} u^n) h(\frac{u}{x}) \log ^r \big (\frac{u}{x}\big ) \frac{du}{u}\\=: & {} I_1 + I_2 + \cdots + I_r + R. \end{aligned}$$

For every \(j= 1, 2, \ldots , r\) we have, with the change of variable \(e^{-k} u^n= t,\)

$$\begin{aligned} I_j= & {} \sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) n \int _0^\infty \psi (e^{-k} u^n) \frac{(\varTheta ^j f)(x)}{j!} \log ^j \big (\frac{u}{x}\big ) \frac{du}{u}\\= & {} \frac{(\varTheta ^j f)(x)}{j!} \sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) \int _0^\infty \psi (t) \bigg ( \frac{1}{n} (\log t +k) - \log x \bigg )^j \frac{dt}{t}\\= & {} \frac{(\varTheta ^j f)(x)}{j!\, n^j} \sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) \int _0^\infty \psi (t) (\log t + k - \log x^n) ^j \frac{dt}{t}\\= & {} \frac{(\varTheta ^j f)(x)}{j! \, n^j} \sum _{\nu =0}^j \left( \begin{array}{c} j \\ \nu \end{array} \right) \sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) (k- \log x^n)^{j-\nu } \int _0^\infty \psi (t) \log ^\nu t \frac{dt}{t} \\= & {} \frac{(\varTheta ^j f)(x)}{j! \, n^j} \sum _{\nu =0}^j \left( \begin{array}{c} j \\ \nu \end{array} \right) m_{j- \nu }(\varphi ) \widetilde{m}_\nu (\psi ). \end{aligned}$$

Now we evaluate the term

$$\begin{aligned} R = \sum _{k = -\infty }^\infty \varphi (e^{-k} x^n) n \int _0^\infty \psi (e^{-k} u^n) h(\frac{u}{x}) \log ^r \big (\frac{u}{x}\big ) \frac{du}{u}. \end{aligned}$$

Let \(\varepsilon >0\) be fixed and let \(\delta\) be such that \(|h(t)| < \varepsilon\) whenever \(|\log t|< \delta .\) So we have

$$\begin{aligned} |R|\le & {} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| n \bigg \{ \int _{|\log (u/x)|<\delta } + \int _{|\log (u/x)|\ge \delta } \bigg \} |\psi (e^{-k} u^n) | |h(\frac{u}{x}) | |\log ^r \big (\frac{u}{x}\big ) |\frac{du}{u} \\=: & {} R_1 + R_2. \end{aligned}$$

For \(R_1\) we have, by the same change of variable as before,

$$\begin{aligned} R_1\le & {} \varepsilon \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| n \int _{|\log (u/x)|<\delta } |\psi (e^{-k} u^n) | |\log ^r \frac{u}{x} |\frac{du}{u} \\= & {} \varepsilon \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _{|\log (\frac{t^{1/n}e^{k/n}}{x})|< \delta } |\psi (t)| \big |\log ^r (\frac{t^{1/n} e^{k/n}}{x})\big | \frac{dt}{t}\\= & {} \frac{\varepsilon }{n^r} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _{|\log (\frac{t^{1/n}e^{k/n}}{x})|< \delta } |\psi (t)| | \log t +k - \log x^n|^r \frac{dt}{t}. \end{aligned}$$

Since \(|\log t +k - \log x^n| ^r\le 2^{r-1}(|\log t|^r + |k - \log x^n|^r)\) we have

$$\begin{aligned} R_1 \le \frac{\varepsilon 2^{r-1}}{n^r} \big ( M_0(\varphi ) \widetilde{M}_r(\psi ) + \widetilde{M}_0(\psi ) M_r(\varphi ) \big ). \end{aligned}$$

For \(R_2\) we have

$$\begin{aligned} R_2&\le \frac{\Vert h\Vert _\infty }{n^r} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _{|\log (\frac{t e^{k}}{x^n})^{1/n}| \ge \delta } |\psi (t)| |\log t + k - \log x^n |^r \frac{dt}{t}\\ &\le \frac{2^{r-1}\Vert h\Vert _\infty }{n^r} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _0^{ e^{-n \delta } x^n e^{-k}} |\psi (t)| | k - \log x^n |^r \frac{dt}{t}\\ & {\quad} +\frac{2^{r-1}\Vert h\Vert _\infty }{n^r} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _{ e^{n \delta } x^n e^{-k}}^\infty |\psi (t)| | k - \log x^n |^r \frac{dt}{t}\\& {\quad} + \frac{ 2^{r-1}\Vert h\Vert _\infty }{n^r} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _{ e^{n \delta } x^n e^{-k}}^\infty |\psi (t)| |\log t |^r \frac{dt}{t} \\& {\quad} + \frac{ 2^{r-1}\Vert h\Vert _\infty }{n^r} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| \int _0^{ e^{-n \delta } x^n e^{-k}} |\psi (t)| |\log t |^r \frac{dt}{t} \\=: & {} R^1_2 + R^2_2 + R^3_2 + R^4_2. \end{aligned}$$

We consider only the term \(R^1_2,\) since the other terms can be estimated in a similar way. We have

$$\begin{aligned} R^1_2= & {} \frac{2^{r-1}\Vert h\Vert _\infty }{n^r} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n)| | k - \log x^n |^r \int _0^{ e^{-n \delta } x^n e^{-k}} |\psi (t)| \frac{dt}{t} = \frac{2^{r-1}\Vert h\Vert _\infty }{n^r} \cdot \\ &\cdot \bigg ( \sum _{ |\log (e^{-k}x^n)|< n\delta /2} +\sum _{ |\log (e^{-k}x^n)| \ge n \delta /2} \bigg ) |\varphi (e^{-k} x^n)| | k - \log x^n |^r \int _0^{ e^{-n \delta } x^n e^{-k}} |\psi (t)| \frac{dt}{t}\\ =: & {} R^{1,1}_2 + R^{1,2}_2. \end{aligned}$$

For \(R^{1,1}_2\) since \(|\log (e^{-k}x^n)|< n\delta /2\) in an analogous way to Theorem 1 we can obtain

$$\begin{aligned} \int _0^{ e^{-n \delta } x^n e^{-k}} |\psi (t)| \frac{dt}{t}\le \varepsilon \end{aligned}$$

and so

$$\begin{aligned} R^{1,1}_2 \le \frac{2^{r-1}\Vert h\Vert _\infty }{n^r} \varepsilon M_r (\varphi ). \end{aligned}$$

For \(R^{1,2}_2\) by assumption (\(\varphi .3\)) and for large n we have

$$\begin{aligned} R^{1,2}_2 \le \frac{2^{r-1}\Vert h\Vert _\infty }{n^r} \varepsilon M_0(\psi ). \end{aligned}$$

Corollary 1

Under the assumptions of Theorem 3 if moreover the functions \(\varphi\) and \(\psi\) are such that \(m_i(\varphi ) = \widetilde{m}_i(\psi ) =0\) for \(i= 1, \ldots , r-1\) then we have

$$\begin{aligned} \lim _{n \rightarrow \infty } n^r [(S^{\varphi , \psi }_n f) (x) - f(x)] = \frac{(\varTheta ^r f)(x)}{r! } (m_r(\varphi ) + \widetilde{m}_r(\psi )). \end{aligned}$$

6 A quantitative estimate of the convergence

Here we state a quantitative approximation result for functions \(f\in \mathcal {C}({\mathbb {R}}^+)\) in terms of the following modulus of continuity

$$\begin{aligned} \omega (f, \delta ) := \sup \{|f(u) - f(v) | : |\log u - \log v| \le \delta \}, \quad \delta >0. \end{aligned}$$

Note that \(\omega\) satisfies all the classical properties of a modulus of continuity. In particular we will employ the following one:

$$\begin{aligned} \omega (f, \lambda \delta ) \le (\lambda + 1) \omega (f, \delta ), \end{aligned}$$

for every \(\delta , \lambda >0.\)

We have the following quantitative estimate

Theorem 4

Let \(\varphi \in \varPhi , \psi \in \Psi\) be such that \(M_1(\varphi )\) and \(\widetilde{M}_1(\psi )\) are finite. If \(f\in \mathcal {C}({\mathbb {R}}^+),\) then for every \(\delta >0\) we have

$$\begin{aligned} | (S^{\varphi , \psi }_n f) (x) - f(x) | \le \omega (f, \delta ) M_0(\varphi ) \widetilde{M}_0(\psi ) + \frac{\omega (f, \delta )}{n \delta } (M_0(\varphi ) \widetilde{M}_1(\psi ) + M_1(\varphi ) \widetilde{M}_0(\psi ) ). \end{aligned}$$

Proof

We have

$$\begin{aligned} | (S^{\varphi , \psi }_n f) (x) - f(x) |\le & {} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n) |n \int _0^\infty |\psi (e^{-k} u^n)| |f(u) - f(x) | \frac{du}{u}\\\le & {} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n) |n \int _0^\infty |\psi (e^{-k} u^n)| \omega (f, |\log (\frac{u}{x}) |) \frac{du}{u}. \end{aligned}$$

Then for any \(\delta >0,\) using the properties of the modulus \(\omega ,\) we obtain

$$\begin{aligned}&| (S^{\varphi , \psi }_n f) (x) - f(x) | \\ & {\quad}\le \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n) |n \int _0^\infty |\psi (e^{-k} u^n)| \bigg ( 1 + \frac{|\log (u/x)|}{\delta } \bigg ) \omega (f, \delta ) \frac{du}{u}\\& {\quad}\le \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n) |n \int _0^\infty |\psi (e^{-k} u^n)| \omega (f, \delta ) \frac{du}{u}\\& {\qquad}+ \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n) |n \int _0^\infty |\psi (e^{-k} u^n)| \frac{|\log (u/x)|}{\delta } \omega (f, \delta ) \frac{du}{u}\\ & {\quad} \le \omega (f, \delta ) M_0(\varphi ) \widetilde{M}_0(\psi ) \\ & {\qquad} + \frac{\omega (f, \delta )}{\delta } \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n) |n \int _0^\infty |\psi (e^{-k} u^n)| |\log (u/x)| \frac{du}{u}. \end{aligned}$$

Using the change of variable \(e^{-k} u^n =t\) we have for the last series

$$\begin{aligned}&\sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n) |n \int _0^\infty |\psi (e^{-k} u^n)| |\log (u/x)| \frac{du}{u} \\= & {\quad} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n) | \int _0^\infty |\psi (t) ||\log (\frac{te^k}{x^n} )^{1/n}| \frac{dt}{t}\\\le & {\quad} \frac{1}{n} \sum _{k = -\infty }^\infty |\varphi (e^{-k} x^n) | \int _0^\infty |\psi (t)| ( |\log t| + |k - \log x^n| ) \frac{dt}{t}\\ \le & {\quad} \frac{1}{n} (M_0(\varphi ) \widetilde{M}_1(\psi ) + M_1(\varphi ) \widetilde{M}_0(\psi ) ), \end{aligned}$$

and so the assertion follows.

As a consequence of the previous theorem we have

Corollary 2

Under the assumption of Theorem 4 we obtain

$$\begin{aligned} | (S^{\varphi , \psi }_n f) (x) - f(x) | \le A \omega (f, \frac{1}{n}), \end{aligned}$$

with A an absolute constant depending only on \(\varphi\) and \(\psi .\)

Proof

It is sufficient to set in Theorem 4\(\delta = \dfrac{1}{n}\) for any fixed n. Then the assertion follows with \(A= M_0(\varphi ) \widetilde{M}_0(\psi ) +M_1(\varphi ) \widetilde{M}_0(\psi ) + M_0(\varphi ) \widetilde{M}_1(\psi ).\)

7 Some examples

In this section we will apply the previous theory to various specific examples. We recall that the Mellin transform of a function \(f:{\mathbb {R}}^+ \rightarrow {\mathbb {R}}\) is defined by (see e.g. [18, 28])

$$\begin{aligned}{}[f]^\wedge _M(s) := \int _0^\infty x^{s-1} f(x) dx, \quad (s= it, t\in {\mathbb {R}}), \end{aligned}$$

whenever the function f is such that

$$\begin{aligned} \int _0^\infty |f(x)| \frac{dx}{x} < \infty . \end{aligned}$$
  1. 1)

    We begin with an important class of functions with compact support, which represents the analogue in the Mellin setting of the classical central B-splines. For every fixed \(n \in {\mathbb {N}}\) we define

    $$\begin{aligned} B_n(x)= \frac{1}{(n-1)!}\sum _{j=0}^n (-1)^j\left( \begin{array}{c} n\\ j\end{array} \right) \bigg (\frac{n}{2} + \log x - j\bigg )_{+}^{n-1}\qquad (x \in {{\mathbb {R}}^+}), \end{aligned}$$

    where, for every \(r \in {\mathbb {R}},\) \(r_+\) denotes the positive part of the number r. Let us consider the second order Mellin spline defined by

    $$\begin{aligned} B_2(x) := (1-|\log x|)_{+}= \left\{ \begin{array}{lll} 1-\log x, &{} \quad 1<x<e\\ 1+\log x, &{} \quad e^{-1}< x < 1\\ 0, &{} \quad \text{ otherwise } \end{array} \right. \end{aligned}$$

    We put \(\varphi (x) = \psi (x) = B_2(x).\) As to the moments of \(\psi\) we have that

    $$\begin{aligned} \widetilde{m}_0(B_2)&= \widetilde{M}_0(B_2) = \int _{e^{-1}}^ 1 (1 + \log x) \frac{dx}{x} + \int _1^e (1- \log x) \frac{dx}{x} = 1\\ \widetilde{m}_1(B_2)&= 0 \quad \widetilde{M}_1(B_2) = \frac{1}{3}. \end{aligned}$$

    For the moments of \(\varphi\) since the function \(B_2\) has compact support, all the absolute moments \(M_j(B_2)\) are finite. Indeed the series have a finite numbers of non-zero terms. 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

    $$\begin{aligned} m_0(B_2) = \sum _{k=-\infty }^\infty B_2(e^{-k} x) = 1, \quad m_1(B_2) = 0. \end{aligned}$$

    In this case the asymptotic formula reduces to

    $$\begin{aligned} \lim _{n \rightarrow \infty } n [(S^{ B_2, B_2}_n f) (x) - f(x)] = 0. \end{aligned}$$
  2. 2)

    Let us consider the generalized Mellin–Jackson kernel, which is defined by (see [12])

    $$\begin{aligned} J_{\gamma , \beta }(x) := d_{\gamma , \beta }~ \text{ sinc}^{2\beta }\bigg (\frac{\log x}{2\gamma \beta \pi }\bigg ), \end{aligned}$$

    where \(x \in {\mathbb {R}}^+,~\beta \in {\mathbb {N}},~\gamma \ge 1,~d_{\gamma , \beta }\) is a normalization constant, i.e.

    $$\begin{aligned} d_{\gamma , \beta }^{-1}:=\int _0^{+\infty }\text{ sinc}^{2\beta }\bigg (\frac{\log x}{2\gamma \beta \pi }\bigg )\frac{du}{u}. \end{aligned}$$

    It is known that \([J_{\gamma , \beta }]^\wedge _M(iv) =0\) for \(|v| \ge 1/\gamma ,\) thus \(J_{\gamma , \beta }\) is Mellin band-limited. We put \(\varphi = J_{\gamma , \beta }(x).\) Using the Mellin–Poisson summation formula (see [9]) we obtain

    $$\begin{aligned} \sum _{k=-\infty }^\infty J_{\gamma , \beta }(e^kx) = \sum _{k=-\infty }^\infty [J_{\gamma , \beta }]^\wedge _M(2k\pi i)x^{-2k\pi i} = [J_{\gamma , \beta }]^\wedge _M(0) = 1. \end{aligned}$$

    So we can prove that assumptions (\(\varphi .1\)), (\(\varphi .2\)) and (\(\varphi .3\)) are satisfied, see [9]. Concerning the moments, using again the Mellin–Poisson summation formula for the derivatives as in [9], one has \(m_1(J_{\gamma , \beta }) = 0,\) and, for \(\beta >3/2,\) (see [12])

    $$\begin{aligned} m_2(J_{\gamma , \beta }) = d_{\gamma , \beta }\int _0^{+\infty } \text{ sinc}^{2\beta }\bigg (\frac{\log x}{2\gamma \beta \pi }\bigg ) \log ^2x\frac{dx}{x}=: A_{\gamma , \beta } < +\infty . \end{aligned}$$

    Moreover, \(M_2(J_{\gamma , \beta }) < +\infty .\) For the function \(\psi\) we put \(\psi (x) = B_2(x)\) the previous spline. We have that

    $$\begin{aligned} \widetilde{m}_2(B_2) = \widetilde{M}_2(B_2) = \frac{1}{6}. \end{aligned}$$

    Therefore the assumptions of the previous theorems are satisfied, with r = 2. In particular we obtain the following Voronovskaja formula, for \(f \in C({\mathbb {R}}^+)\) of class \(C^{(2)}\) locally at the point \(x \in {\mathbb {R}}^+:\)

    $$\begin{aligned} \lim _{n \rightarrow +\infty }n^2[(S_n^{J_{\gamma , \beta }, B_2} f)(x) - f(x)] = \frac{\varTheta ^2 f(x)}{2} (A_{\gamma , \beta } + \frac{1}{6}). \end{aligned}$$