1 Introduction

Bernstein polynomials are useful tool to prove the well-known Weierstrass approximation theorem for the space of continuous functions on \(\left[ 0,1\right] \) or more generally on \(\left[ a,b\right] \subset \mathbb {R}\) (see [15]). In [32], King constructed and studied a generalization of the classical Bernstein operators using a sequence of continuous functions defined on [0, 1], \(n\in \mathbb {N}\) to obtain a better approximation. In [23], the authors introduced a new type of operators \(B_n\) in the form

$$\begin{aligned} B_n^\rho \left( f;x\right) :=\sum _{k=0}^{n}\left( f\circ \rho ^{-1}\right) \left( \frac{k}{n}\right) {n \atopwithdelims ()k}\rho ^k\left( x\right) \left( 1-\rho \left( x\right) \right) ^{n-k}, \ x\in \left[ 0,1\right] ,n\in \mathbb {N}, \end{aligned}$$

using a special function \(\rho :\left[ 0,1\right] \rightarrow \mathbb {R}\) that satisfies suitable assumptions. In the same paper, the authors obtained that a new family of operators gives a better approach than the operators \(B_n\) in certain cases. Similar constructions have been studied for other sequence of linear positive operators; we refer the readers to [1, 2, 7, 29, 31].

To obtain an approximation over the whole real axis, Butzer and his school (see [16, 19, 21, 22]) introduced the generalized sampling operators given by

$$\begin{aligned} \left( G_w^\chi f\right) \left( x\right) :=\sum _{k\in \mathbb {Z}}f\left( \frac{k}{w}\right) \chi \left( wx-k\right) ,\ x\in \mathbb {R},w>0, \end{aligned}$$
(1.1)

where \(\chi :\mathbb {R}\rightarrow \mathbb {R}\) is called a kernel function satisfying certain assumptions of approximate identities and \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a bounded, continuous function on \(\mathbb {R}\). The generalized sampling operators given in (1.1), were considered as an approximate version of the classical Whittaker–Kotel’nikov–Shannon sampling theorem (see [33, 40, 41]). In recent years, numerous studies have been published on sampling type operators. We can refer the readers to [19, 21, 28, 39] for generalized sampling operators, [9, 11, 26] for sampling Kantorovich operators, [14, 24] for sampling Durrmeyer operators, [5, 6, 12, 37] for exponential sampling type operators and [3, 4, 8, 10, 34,35,36, 38] for both polynomial and logarithmic weighted approximation by sampling type operators.

Our aim in this paper is to construct a new form of generalized sampling operators given in (1.1) by considering a \(\rho \) function, which satisfies some suitable conditions. Such a construction is important when we face signals which are not smoothly spaced, and this means that we can not use the operators (1.1) for these signals. The paper is organized as follows: Sect. 2 is devoted to basic notation and preliminaries. In Sect. 3, we deal with the main approximation properties of the newly constructed operators. In Sect. 4, we give a Voronovskaja-type formula for these operators. Also, we present a comparison theorem between newly constructed and classical generalized sampling operators. In Sect. 5, we study convergence of these operators in weighted spaces of continuous functions by taking a general weight function. Finally, in Sects. 6 and 7, we give some examples of the kernels satisfying suitable assumptions and by considering a special \(\rho \) function, we present some graphical and numerical representations to compare the modified generalized sampling operators \(G_w^{\chi ,\rho }\) and the classical generalized sampling operators \(G_w^\chi \).

2 Basic Notations and Preliminaries

By \(\mathbb {N},\mathbb {Z}\) and \(\mathbb {R}\), we shall denote the sets of all positive integers, integers and real numbers, respectively.

By \(C\left( \mathbb {R}\right) \), we will denote the space of all continuous (not necessarily bounded) functions defined on \(\mathbb {R}\) and by \(CB\left( \mathbb {R}\right) \) the space of all bounded functions \(f\in C\left( \mathbb {R}\right) \) endowed with the norm \(\left\| f \right\| :=\sup _{x\in \mathbb {R}}\left| f\left( x\right) \right| \). Moreover, by \(UC\left( \mathbb {R}\right) \), we denote the subspaces of \(CB\left( \mathbb {R}\right) \) comprising all uniformly continuous functions.

Let \(\rho :\mathbb {R}\rightarrow \mathbb {R}\) be a strictly increasing function that satisfies the following conditions:

\((\rho _1)\):

\(\rho \in C\left( \mathbb {R}\right) \);

\((\rho _2)\):

\(\rho \left( 0\right) =0\), \(\displaystyle \lim _{x\pm \infty }\rho \left( x\right) =\pm \infty \).

Definition 1

Throughout the paper, a function \(\chi :\mathbb {R}\rightarrow \mathbb {R}\) is said to be a kernel associated with \(\rho \) (or simply \(\rho \)-kernel) if it satisfies the following assumptions:

\((\chi 1)\):

\(\chi \in C\left( \mathbb {R}\right) \);

\((\chi 2)\):

for every \(u\in \mathbb {R}\), discrete \(\rho \)-algebraic moment of order 0 of \(\chi \) is 1, that is

$$\begin{aligned} m_0^\rho \left( \chi ,u\right) =\sum _{k\in \mathbb {Z}}\chi \left( \rho \left( u\right) -k\right) =1; \end{aligned}$$
\((\chi 3)\):

for any \(\beta \ge 0\), absolute moment of order \(\beta \) associated with \(\rho \) of \(\chi \) (or simpliy \(\rho \)-absolute moment)   is finite, that is

$$\begin{aligned} M_{\beta }^\rho \left( \chi \right) =\sup _{u\in \mathbb {R}}\sum _{k\in \mathbb {Z}}\left| \chi \left( \rho \left( u\right) -k\right) \right| \left| k-\rho \left( u\right) \right| ^{\beta }<\infty . \end{aligned}$$

By \(\psi \), we will denote the class of all functions satisfying the assumptions \((\chi 1), (\chi 2)\) and \((\chi 3)\).

For any function \(\chi :\mathbb {R}\rightarrow \mathbb {R}\), a discrete algebraic moment of order \(j\in \mathbb {N}\cup \left\{ 0\right\} \) associated with \(\rho \) of \(\chi \) (or simply \(\rho \)-algebraic moment) is defined by

$$\begin{aligned} m_j^\rho \left( \chi ,u\right) :=\sum _{k\in \mathbb {Z}}\chi \left( \rho \left( u\right) -k\right) \left( k-\rho \left( u\right) \right) ^j, \ u\in \mathbb {R}. \end{aligned}$$
$$\begin{aligned} M_\beta ^\rho \left( \chi \right) :=\sup _{u\in \mathbb {R}}\sum _{k\in \mathbb {Z}}\left| \chi \left( \rho \left( u\right) -k\right) \right| \left| k-\rho \left( u\right) \right| ^\beta , \ u\in \mathbb {R}. \end{aligned}$$

Remark 1

  1. (i)

    Let \(\chi \) be a function satisfying \((\chi 1)\) and \((\chi 3)\), there holds:

    $$\begin{aligned} \lim _{w\rightarrow \infty }\sum _{\left| k-w\rho \left( x\right) \right| \ge w\delta }\left| \chi \left( w\rho \left( x\right) -k\right) \right| =0 \end{aligned}$$

    uniformly with respect to \(x\in \mathbb {R}\) (see [11]).

  2. (ii)

    For \(\eta , \gamma > 0\) with \(\eta<\gamma , M_\gamma ^\rho (\chi )<\infty \) implies \(M_\eta ^\rho (\chi )<\infty \). When \(\chi \) has compact support, we immediately have that \(M_\gamma ^\rho (\chi )<\infty \) for every \(\gamma >0\) (see [25]).

Now, we introduce a new family of sampling type operators, so-called modified generalized sampling operators, by

$$\begin{aligned} \left( G_{w}^{\chi ,\rho }f\right) \left( x\right)= & {} \left[ G_{w}^{\chi }\left( f\circ \rho ^{-1}\right) \right] \left( \rho \left( x\right) \right) \nonumber \\:= & {} \sum _{k\in \mathbb {Z}}\left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) \chi \left( w\rho \left( x\right) -k\right) , x\in \mathbb {R},w>0 \end{aligned}$$
(2.1)

for \(\chi \in \psi \).

Remark 2

The operator (2.1) is well-defined if, for example, f is bounded. Indeed, if \(\left| f\left( x\right) \right| \le L\) for every \(x\in \mathbb {R}\), then \(f\circ \rho ^{-1}\) is also a bounded function. Then

$$\begin{aligned} \left| \left( G_w^{\chi ,\rho } f\right) \left( x\right) \right|&\le \sum _{k\in \mathbb {Z}}\left| \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) \right| \left| \chi \left( w\rho \left( x\right) -k\right) \right| \\&\le L\sum _{k\in \mathbb {Z}}\left| \chi \left( w\rho \left( x\right) -k\right) \right| \\&\le LM_0^\rho \left( \chi \right) <\infty . \end{aligned}$$

Remark 3

In the special case of \(\rho \left( x\right) =x\) (it is clear that \((\rho 1)\) and \((\rho 2)\) are satisfied), the operators (2.1) reduce to the classical generalized sampling series

$$\begin{aligned} \left( G_{w}^{\chi } f\right) \left( x\right) =\sum _{k\in \mathbb {Z}}f\left( \frac{k}{w}\right) \chi \left( wx-k\right) . \end{aligned}$$

3 Approximation Results for \(G_{w}^{\chi ,\rho }\)

In this section, we present some approximation results for the family of operators \(\left( G_w^{\chi ,\rho }\right) \) including pointwise convergence, uniform convergence, and rate of convergence.

Theorem 1

Let \(\chi \in \psi \) be a \(\rho \)-kernel. If \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a bounded function, then

$$\begin{aligned} \lim _{w\rightarrow \infty }\left( G_{w}^{\chi ,\rho } f\right) \left( x_0\right) =f\left( x_0\right) \end{aligned}$$
(3.1)

holds at each continuity point \(x_{0} \in \mathbb {R}\) of f.

Proof

Assume that \(x_{0}\) be a continuity point of f. So, \(\left( f\circ \rho ^{-1}\circ \rho \right) \) is also continuous at the point \(x_{0}\) and that means \(f\circ \rho ^{-1}\) is continuous at \(\rho \left( x_{0}\right) \). Hence, for every \(\varepsilon >0\), there exists \(\delta >0\) such that \(\left| \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) - \left( f\circ \rho ^{-1}\right) \left( \rho \left( x_0\right) \right) \right| <\varepsilon \) whenever \(\left| \frac{k}{w}-\rho \left( x_0\right) \right| <\delta \). Thus, we can write

$$\begin{aligned}&\left| \left( G_{w}^{\chi , \rho } f\right) \left( x_0\right) -f\left( x_0\right) \right| \\ {}&\quad \le \sum _{k\in \mathbb {Z}}\left| \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) -\left( f\circ \rho ^{-1}\right) \left( \rho \left( x_0\right) \right) \right| \left| \chi \left( w\rho \left( x_0\right) -k\right) \right| \\ {}&\quad =\left( \sum _{\left| k-w\rho \left( x_0\right) \right| <w\delta }+\sum _{\left| k-w\rho \left( x_0\right) \right| \ge w\delta }\right) \left| \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) \right. \\ {}&\qquad \left. -\left( f\circ \rho ^{-1}\right) \left( \rho \left( x_0\right) \right) \right| \left| \chi \left( w\rho \left( x_0\right) -k\right) \right| \\ {}&\quad =:S_1+S_2. \end{aligned}$$

Let us first consider \(S_1\). Since \(f\circ \rho ^{-1}\) is continuous at \(\rho \left( x_0\right) \), we get

$$\begin{aligned} S_1\le \varepsilon M_0^\rho \left( \chi \right) . \end{aligned}$$

Now we estimate \(S_2\). In view of Remark 1 (i) with \(\beta =0\), we have

$$\begin{aligned} S_2&\le 2\left\| f\circ \rho ^{-1} \right\| _{\infty }\sum _{\left| k-w\rho \left( x_0\right) \right| \ge w\delta }\left| \chi \left( w\rho \left( x_0\right) -k\right) \right| \\ {}&<2\left\| f\circ \rho ^{-1}\right\| _{\infty }\varepsilon \end{aligned}$$

for sufficiently large w. Combining \(S_{1}\) and \(S_{2}\) and taking limit as \(w\rightarrow \infty \) we conclude (3.1). \(\square \)

Theorem 2

Let \(\chi \in \psi \) be a \(\rho \)-kernel. If \(f\circ \rho ^{-1}\in UC\left( \mathbb {R}\right) \), then

$$\begin{aligned} \lim _{w\rightarrow \infty }\left\| G_{w}^{\chi , \rho } f-f\right\| _{\infty }=0 \end{aligned}$$

holds.

Proof

The proof follows the same argument as Theorem 1, taking into account that if \(f\circ \rho ^{-1}\in UC\left( \mathbb {R}\right) \), then we can choose \(\delta >0\) independent of x such that for \(\left| k-w\rho \left( x\right) \right| <w\delta \), one has \(\left| \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) -\left( f\circ \rho ^{-1}\right) \left( \rho \left( x\right) \right) \right| <\varepsilon \) uniformly with respect to \(x\in \mathbb {R}\). \(\square \)

Remark 4

We can not change the assumption \(f\circ \rho ^{-1}\in UC\left( \mathbb {R}\right) \) to \(f\in UC\left( \mathbb {R}\right) \) in Theorem 2, since uniform continuity of f does not guarantee uniform continuity of \(f\circ \rho ^{-1}\). For example, consider \(f:\mathbb {R}\rightarrow \mathbb {R}\), \(f\left( x\right) = x\) and \(\rho \left( x\right) = \root 3 \of {x}\).

Now, we will give a quantitative estimate for functions \(f\in CB\left( \mathbb {R}\right) \) via the classical modulus of continuity. First, let us remind the definition of the modulus of continuity. For functions \(f\in CB\left( \mathbb {R}\right) \) and \(\delta >0\), the modulus of continuity is defined by

$$\begin{aligned} \omega \left( f, \delta \right) = \sup _{\begin{array}{c} t, x\in \mathbb {R}\\ \left| t-x\right| <\delta \end{array}} \left| f\left( t\right) - f\left( x\right) \right| . \end{aligned}$$

The modulus of continuity satisfies the following properties:

For every \(f\in CB\left( \mathbb {R}\right) \) and any \(\lambda >0\),

$$\begin{aligned} \omega \left( f, \lambda \delta \right) \le \left( 1+\lambda \right) \omega \left( f, \delta \right) \end{aligned}$$
(3.2)

and moreover, if \(f\in UC\left( \mathbb {R}\right) \), then

$$\begin{aligned} \lim \limits _{\delta \rightarrow 0}\omega \left( f,\delta \right) =0 \end{aligned}$$
(3.3)

holds (see [27]).

Theorem 3

Let \(f\in CB\left( \mathbb {R}\right) \). If \(\chi \in \psi \) be a \(\rho \)-kernel with \(M_1^\rho \left( \chi \right) <\infty \), then we have

$$\begin{aligned} \left| \left( G_{w}^{\chi ,\rho } f\right) \left( x\right) -f\left( x\right) \right| \le \omega \left( f\circ \rho ^{-1},w^{-1}\right) \left( M_0^\rho \left( \chi \right) +M_1^\rho \left( \chi \right) \right) . \end{aligned}$$
(3.4)

Proof

Using the definition of the operators and (3.2), we have by direct computation that

$$\begin{aligned} \left| \left( G_{w}^{\chi ,\rho } f\right) \left( x\right) -f\left( x\right) \right|&\le \sum _{k\in \mathbb {Z}}\omega \left( f\circ \rho ^{-1},\left| \frac{k}{w}-\rho \left( x\right) \right| \right) \left| \chi \left( w\rho \left( x\right) -k\right) \right| \\&\le \sum _{k\in \mathbb {Z}}\left( 1+\frac{\left| k-w\rho \left( x\right) \right| }{w\delta }\right) \omega \left( f\circ \rho ^{-1},\delta \right) \left| \chi \left( w\rho \left( x\right) -k\right) \right| \end{aligned}$$

and choosing \(\delta =w^{-1}\), we get the desired result. \(\square \)

4 Voronovskaja-Type Formula

In this section, we give a qualitative form of the Voronovskaja-type formula by using Taylor expansion. Additionally, we need more assumptions on functions \(\chi \) and \(\rho \) to state and prove the Voronovskaja-type theorem:

\((\rho _3)\):

Let \(\rho \) be a continuously differentiable function

and

\((\chi 4)\):

\(M_{1}^{\rho }\left( \chi \right) <\infty \) and

$$\begin{aligned} \lim _{w\rightarrow \infty }\sum _{\left| k-w\rho \left( x\right) \right| \ge w\delta }\left| \chi \left( w\rho \left( x\right) -k\right) \right| \left| k - w\rho \left( x\right) \right| =0 \end{aligned}$$

holds uniformly with respect to \(x\in \mathbb {R}\).

There are many kernels that satisfy the assumption \((\chi 4)\); for instance, Translates of B-splines, Bochner–Riesz kernel, generalized Jackson kernel, for details, see [13].

Theorem 4

Let \(f\in CB\left( \mathbb {R}\right) \). Suppose that \(f'\) and \(\rho '\) exist at any \(x\in \mathbb {R}\) and \(m_{1}^{\rho }\left( \chi ,x\right) := m_{1}^\rho \left( \chi \right) \ne 0\) is independent of x. If \(\chi \in \psi \) be a \(\rho \)-kernel such that \((\chi 4)\) is satisfied, then we have

$$\begin{aligned} \lim \limits _{w\rightarrow \infty }w\left[ \left( G_{w}^{\chi , \rho } f\right) \left( x\right) -f\left( x\right) \right] = \dfrac{f'\left( x\right) }{\rho '\left( x\right) }m_{1}^{\rho }\left( \chi \right) +o\left( w^{-1}\right) . \end{aligned}$$

Proof

By the Taylor expansion of \(f\circ \rho ^{-1}\) at the point \(\rho \left( x\right) \in \mathbb {R}\), we have

$$\begin{aligned} \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right)&=f\left( x\right) +\left( f\circ \rho ^{-1}\right) ^{\prime }\left( \rho \left( x\right) \right) \left( \frac{k}{w}-\rho \left( x\right) \right) +h\left( \frac{k}{w}\right) \left( \frac{k}{w}-\rho \left( x\right) \right) , \nonumber \\ \end{aligned}$$
(4.1)

where h is a bounded function such that

$$\begin{aligned} \lim \limits _{\frac{k}{w}\rightarrow \rho \left( x\right) }h\left( \frac{k}{w}\right) = 0. \end{aligned}$$
(4.2)

Now, using the definition of the operators (2.1) and the equality (4.1), we get

$$\begin{aligned}&\left( G_{w}^{\chi , \rho } f\right) \left( x\right) -f\left( x\right) \\&\quad =\sum _{k\in \mathbb {Z}}\chi \left( w\rho \left( x\right) -k\right) \left[ \dfrac{f^{\prime }\left( x\right) }{\rho ^{\prime }\left( x\right) }\left( \frac{k}{w}-\rho \left( x\right) \right) + h\left( \frac{k}{w}\right) \left( \frac{k}{w}-\rho \left( x\right) \right) \right] \\&\quad =\sum _{k\in \mathbb {Z}}\chi \left( w\rho \left( x\right) -k\right) \dfrac{f^{\prime }\left( x\right) }{\rho ^{\prime }\left( x\right) }\left( \frac{k}{w}-\rho \left( x\right) \right) \\&\qquad + \sum _{k\in \mathbb {Z}}\chi \left( w\rho \left( x\right) -k\right) h\left( \frac{k}{w}\right) \left( \frac{k}{w}-\rho \left( x\right) \right) \\&\quad =:I_{1}+I_{2}. \end{aligned}$$

Indeed, it is easy to see that

$$\begin{aligned} I_{1} = \dfrac{f^{\prime }\left( x\right) }{\rho ^{\prime }\left( x\right) }\dfrac{1}{w}m_{1}^\rho \left( \chi \right) . \end{aligned}$$

Now, let us estimate \(I_{2}\). We write

$$\begin{aligned} \left| I_{2}\right|&= \left( \sum _{\left| k-w\rho \left( x\right) \right| <w\delta } + \sum _{\left| k-w\rho \left( x\right) \right| \ge w\delta }\right) \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left| h\left( \frac{k}{w}\right) \right| \left| \frac{k}{w} - \rho \left( x\right) \right| \\&:= I_{2,1} + I_{2,2}. \end{aligned}$$

Using (4.2), we have that \(I_{2,1} \le \frac{\varepsilon }{w}M_{1}^\rho \left( \chi \right) \). Moreover, by using boundedness of h, we have that \(I_{2,2} \le \frac{\left\| h \right\| _{\infty }}{w}\varepsilon \) for sufficiently large w by Remark 1 (i) with \(\beta =1\). Hence, we conclude that

$$\begin{aligned} w\left[ \left( G_{w}^{\chi , \rho } f\right) \left( x\right) -f\left( x\right) \right] = \dfrac{f'\left( x\right) }{\rho '\left( x\right) }m_{1}^{\rho }\left( \chi \right) + \frac{\varepsilon }{w}\left( M_{1}^\rho \left( \chi \right) + \left\| h\right\| _{\infty }\right) \end{aligned}$$

and the assertion follows as \(w\rightarrow \infty \). \(\square \)

Using the similar methods applied in the proof of Theorem 4, the following Corollary can be proved:

Corollary 1

Let \(f\in CB\left( \mathbb {R}\right) \). Suppose that \(f''\) and \(\rho ''\) exist at any \(x\in \mathbb {R}\) and \(m_{2}^{\rho }\left( \chi ,x\right) :=m_{2}^\rho \left( \chi \right) \ne 0\) is independent of x. Suppose also that \(m_{1}^\rho \left( \chi \right) \) is independent of x and \(m_{1}^\rho \left( \chi \right) = 0\). If \(\chi \in \psi \) be a \(\rho \)-kernel such that \((\chi 4)\) is satisfied, then we have

$$\begin{aligned} \lim \limits _{w\rightarrow \infty } w^2\left[ \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) - f\left( x\right) \right] = \left( \dfrac{f''\left( x\right) }{\left[ \rho '\left( x\right) \right] ^2} - \dfrac{f'\left( x\right) \rho ''\left( x\right) }{\left[ \rho '\left( x\right) \right] ^3}\right) m_{2}^\rho \left( \chi \right) . \end{aligned}$$
(4.3)

Theorem 5

Let \(f\in CB\left( \mathbb {R}\right) \). Suppose that \(f'', \rho ''\) exists at any \(x\in \mathbb {R}, m_{1}^{\rho }\left( \chi \right) =0, m_{2}^{\rho }\left( \chi ,x\right) :=m_{2}^{\rho }\left( \chi \right) > 0\) is independent of x, \(\chi \in \psi \) is a \(\rho \)-kernel such that \((\chi 4)\) is satisfied. Assume that there exists \(w'>0\) such that

$$\begin{aligned} f(x)\le \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) \le \left( G_{w}^{\chi }f\right) \left( x\right) \end{aligned}$$
(4.4)

at any point \(x\in \mathbb {R}\) for all \(w>w'\). Then,

$$\begin{aligned} f''\left( x\right) \ge \dfrac{\rho ''\left( x\right) }{\rho '\left( x\right) }f'\left( x\right) \ge \left( 1- \left[ \rho '\left( x\right) \right] ^2\right) f''\left( x\right) , \quad x\in \mathbb {R}. \end{aligned}$$
(4.5)

Conversely, if (4.5) holds with strict inequalities at a given point \({x\in \mathbb {R}}\), then there exists \(w'>0\) such that \(w>w'\)

$$\begin{aligned} f\left( {x}\right)< \left( G_{w}^{\chi ,\rho }f\right) \left( {x}\right) < \left( G_{w}^{\chi }f\right) \left( {x}\right) \end{aligned}$$
(4.6)

for \(w>w'\).

Proof

By the assumption (4.4), we have the inequality

$$\begin{aligned} 0 \le w^2\left[ \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) - f\left( x\right) \right] \le w^2\left[ \left( G_{w}^{\chi }f\right) \left( x\right) - f\left( x\right) \right] \end{aligned}$$

at any point \(x\in \mathbb {R}\) for all \(w>w'\). Then, using (4.3) (recall the classical Voronovskaja theorem for \(G_w^{\chi }\) by the fact that \(\rho \left( x\right) =x\) in (4.3)) we have

$$\begin{aligned} 0 \le \left( \dfrac{f''\left( x\right) }{\left[ \rho '\left( x\right) \right] ^2} - \dfrac{f'\left( x\right) \rho ''\left( x\right) }{\left[ \rho '\left( x\right) \right] ^3}\right) \le f''\left( x\right) \end{aligned}$$

which yields (4.5).

Conversely, if (4.5) holds with strict inequalities at any \({x\in \mathbb {R}}\), then multiplying each terms of inequality (4.5) by \(\dfrac{1}{\left[ \rho '\left( x\right) \right] ^2}\) and subtract \(\dfrac{f''\left( x\right) }{\left[ \rho '\left( x\right) \right] ^2}\) from each terms, respectively, we have

$$\begin{aligned} 0< \left( \dfrac{f''\left( x\right) }{\left[ \rho '\left( x\right) \right] ^2} - \dfrac{f'\left( x\right) \rho ''\left( x\right) }{\left[ \rho '\left( x\right) \right] ^3}\right) < f''\left( x\right) \end{aligned}$$

and using again (4.3), desired result is obtained. \(\square \)

Example 1

Let us consider a function \(f:\mathbb {R}\rightarrow \mathbb {R}\) given by \(f(x)=\frac{x^3}{3}\) and \(\rho \left( x\right) = x^3+x\). Under these considerations inequality (4.5) holds for strict inequalities for all \(x\in \mathbb {R}\backslash \{0\}\). So we can say, theoretically, that modified generalized sampling series gives a better approach than classical one for all \(x\in \mathbb {R}\backslash \{0\}\).

5 Weighted Approximation

In this section, we study approximation properties of the modified generalized sampling operators in weighted spaces of continuous functions. Throughout the paper, for the weight function \(\varphi :\mathbb {R}\rightarrow \mathbb {R}, \ \varphi (x)=1+\rho ^2(x)\), we shall consider the following class of functions:

$$\begin{aligned} B_{\varphi }\left( \mathbb {R}\right)&=\left\{ f: \mathbb {R} \rightarrow \mathbb {R} \mid \text { for every } x\in \mathbb {R}, \frac{\left| f\left( x\right) \right| }{\varphi (x)} \le M_f \right\} , \\ C_{\varphi }\left( \mathbb {R}\right)&=C\left( \mathbb {R}\right) \cap B_\varphi \left( \mathbb {R}\right) , \\ U_{\varphi }\left( \mathbb {R}\right)&=\left\{ f \in C_{\varphi }\left( \mathbb {R}\right) \mid \frac{\left| f\left( x\right) \right| }{\varphi \left( x\right) } \text{ is } \text{ uniformly } \text{ continuous } \text{ on } \mathbb {R}\right\} , \end{aligned}$$

where \(M_f\) is a constant depending only on f and the above spaces are normed linear spaces with the norm \(\Vert f\Vert _{\varphi }=\displaystyle \sup _{x \in \mathbb {R}}\frac{\left| f\left( x\right) \right| }{\varphi \left( x\right) }\). The weighted modulus of continuity defined in [30]Footnote 1 is given by

$$\begin{aligned} \omega _\varphi (f ; \delta )=\sup _{\begin{array}{c} x, t \in \mathbb {R} \\ |\rho (t)-\rho (x)| \le \delta \end{array}} \frac{\left| f(t)-f(x)\right| }{\varphi (t)+\varphi (x)} \end{aligned}$$
(5.1)

for each \(f \in C_{\varphi }\left( \mathbb {R}\right) \) and for every \(\delta >0\). We observe that

$$\begin{aligned} \omega _\varphi (f; 0)=0 \end{aligned}$$

for every \(f \in C_{\varphi }\left( \mathbb {R}\right) \) and the function \(\omega _\varphi (f; \delta )\) is nonnegative and nondecreasing with respect to \(\delta \) for \(f \in C_{\varphi }\left( \mathbb {R}\right) \) and also

$$\begin{aligned} \lim \limits _{\delta \rightarrow 0} \omega _\varphi (f ; \delta )=0 \end{aligned}$$
(5.2)

for every \(f \in U_{\varphi }\left( \mathbb {R}\right) \) (for more details, see [30]). We recall the following auxiliary lemma to obtain an estimate for \(\left| f\left( u\right) - f\left( x\right) \right| \).

Lemma 1

([30]) For every \(f \in C_{\varphi }\left( \mathbb {R}\right) \) and \(\delta >0\)

$$\begin{aligned} \left| f(u)-f(x)\right| \le (\varphi (u)+\varphi (x))\left( 2+\frac{\left| \rho (u)-\rho (x)\right| }{\delta }\right) \omega _\varphi (f, \delta ) \end{aligned}$$
(5.3)

holds for all \(x, y \in \mathbb {R}\).

Remark 5

If we consider inequality (5.3), since

$$\begin{aligned} \varphi \left( u\right) +\varphi \left( x\right) \le \delta ^{2} + 2\rho ^{2}\left( x\right) + 2\left| \rho \left( x\right) \right| \delta \text { whenever } \left| \rho \left( u\right) -\rho \left( x\right) \right| \le \delta \end{aligned}$$

and

$$\begin{aligned}{} & {} \varphi \left( u\right) +\varphi \left( x\right) \le \left( \delta ^{2} + 2\rho ^{2}\left( x\right) + 2\left| \rho \left( x\right) \right| \delta \right) \left( \dfrac{\left| \rho \left( u\right) -\rho \left( x\right) \right| }{\delta }\right) ^{2}\\{} & {} \qquad \text { whenever } \left| \rho \left( u\right) -\rho \left( x\right) \right| > \delta , \end{aligned}$$

we get

$$\begin{aligned}&\left| f\left( u\right) -f\left( x\right) \right| \\&\quad \le {\left\{ \begin{array}{ll} \left( \delta ^{2} + 2\rho ^{2}\left( x\right) + 2\left| \rho \left( x\right) \right| \delta +2\right) \\ \qquad \left( 2+ \dfrac{\left| \rho \left( u\right) -\rho \left( x\right) \right| }{\delta }\right) \omega _\varphi \left( f;\delta \right) ,&{} \left| \rho \left( u\right) -\rho \left( x\right) \right| \le \delta \\ \left( \delta ^{2} + 2\rho ^{2}\left( x\right) + 2\left| \rho \left( x\right) \right| \delta +2\right) \left( \dfrac{\left| \rho \left( u\right) -\rho \left( x\right) \right| }{\delta }\right) ^{2}\\ \qquad \left( 2+ \dfrac{\left| \rho \left( u\right) -\rho \left( x\right) \right| }{\delta }\right) \omega _\varphi \left( f;\delta \right) ,&{}\left| \rho \left( u\right) -\rho \left( x\right) \right|> \delta \end{array}\right. }\\&\quad \le {\left\{ \begin{array}{ll} 3\left( \delta ^{2} + 2\rho ^{2}\left( x\right) + 2\left| \rho \left( x\right) \right| \delta +2\right) \omega _\varphi \left( f;\delta \right) ,&{} \left| \rho \left( u\right) -\rho \left( x\right) \right| \le \delta \\ 3\left( \delta ^{2} + 2\rho ^{2}\left( x\right) + 2\left| \rho \left( x\right) \right| \delta +2\right) \dfrac{\left| \rho \left( u\right) -\rho \left( x\right) \right| ^{3}}{\delta ^{3}}\omega _\varphi \left( f;\delta \right) ,&{}\left| \rho \left( u\right) -\rho \left( x\right) \right| > \delta \end{array}\right. }. \end{aligned}$$

If we combine two cases of \(\left| \rho \left( u\right) -\rho \left( x\right) \right| \) with respect to \(\delta \), it turns out that

$$\begin{aligned} \left| f\left( u\right) -f\left( x\right) \right| \le 3\left( \delta ^{2} + 2\rho ^{2}\left( x\right) + 2\left| \rho \left( x\right) \right| \delta +2\right) \omega _\varphi \left( f;\delta \right) \left( 1 + \dfrac{\left| \rho \left( u\right) -\rho \left( x\right) \right| ^{3}}{\delta ^{3}}\right) . \end{aligned}$$

Hence, choosing \(\delta \le 1\), we obtain

$$\begin{aligned} \left| f\left( u\right) -f\left( x\right) \right| \le 9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2 \omega _\varphi \left( f;\delta \right) \left( 1 + \dfrac{\left| \rho \left( u\right) -\rho \left( x\right) \right| ^{3}}{\delta ^{3}}\right) . \end{aligned}$$
(5.4)

As a first main result of this section, we present the well-definiteness of the family of operators \(\left( G_{w}^{\chi ,\rho }\right) \) in weighted spaces of functions.

Theorem 6

Let \(\chi \in \psi \) be a \(\rho \)-kernel with \(M_2^\rho \left( \chi \right) <\infty \). Then, for a fixed \(w>0\), the operator \(G_{w}^{\chi ,\rho }\) is a linear operator from \(B_{\varphi }\left( \mathbb {R}\right) \) to \(B_{\varphi }\left( \mathbb {R}\right) \) and its operator norm turns out to be:

$$\begin{aligned} \left\| G_{w}^{\chi ,\rho }\right\| _{B_\varphi \rightarrow B_\varphi } \le M_{0}^{\rho }\left( \chi \right) + \frac{1}{w^{2}} M_{2}^{\rho }\left( \chi \right) + \frac{2}{w}M_{1}^{\rho }\left( \chi \right) . \end{aligned}$$

Proof

For a fixed \(w>0\) and \(x\in \mathbb {R}\), using the definition of the operators \(G_{w}^{\chi ,\rho }\), we can write

$$\begin{aligned} \left| \left( G_{w}^{\chi ,\rho } f\right) \left( x\right) \right|&\le \sum _{k\in \mathbb {Z}}\left| \chi \left( w\rho \left( x\right) -k\right) \right| \frac{\left| f\left( \rho ^{-1}\left( \frac{k}{w}\right) \right) \right| }{1 + \rho ^{2}\left( \rho ^{-1}\left( \frac{k}{w}\right) \right) }\left[ 1 + \rho ^{2}\left( \rho ^{-1}\left( \frac{k}{w}\right) \right) \right] \\&\le \left\| f \right\| _{\varphi } \sum _{k\in \mathbb {Z}}\left| \chi \left( w\rho \left( x\right) -k\right) \right| \left[ 1 + \left( \frac{k}{w}\right) ^{2}\right] \\&= \left\| f \right\| _{\varphi } \sum _{k\in \mathbb {Z}}\left| \chi \left( w\rho \left( x\right) -k\right) \right| \left[ 1 + \left( \frac{k}{w} - \rho \left( x\right) \right) ^{2}\right. \\&\left. +2\rho \left( x\right) \left( \frac{k}{w}-\rho \left( x\right) \right) +\rho ^{2}\left( x\right) \right] \\&\le \left\| f \right\| _{\varphi }\left( 1 + \rho ^{2}\left( x\right) \right) \sum _{k\in \mathbb {Z}}\left| \chi \left( w\rho \left( x\right) -k\right) \right| \left[ 1 + \frac{1}{w^{2}}\left( k-w\rho \left( x\right) \right) ^{2} \right. \\&\left. + \frac{2}{w}\left| k-w\rho \left( x\right) \right| \right] \\&\le \left\| f \right\| _{\varphi }\left( 1 + \rho ^{2}\left( x\right) \right) \left[ M_{0}^{\rho }\left( \chi \right) + \frac{1}{w^{2}} M_{2}^{\rho }\left( \chi \right) + \frac{2}{w}M_{1}^{\rho }\left( \chi \right) \right] \end{aligned}$$

which implies that

$$\begin{aligned} \frac{\left| \left( G_{w}^{\chi ,\rho } f\right) \left( x\right) \right| }{1 + \rho ^{2}\left( x\right) }\le \left\| f \right\| _{\varphi }\left[ M_{0}^{\rho }\left( \chi \right) + \frac{1}{w^{2}} M_{2}^{\rho }\left( \chi \right) + \frac{2}{w}M_{1}^{\rho }\left( \chi \right) \right] \end{aligned}$$

for every \(x\in \mathbb {R}\) and taking supremum over \(x\in \mathbb {R}\), we have

$$\begin{aligned} \left\| G_{w}^{\chi ,\rho }f \right\| _{\varphi }\le \left\| f \right\| _{\varphi }\left[ M_{0}^{\rho }\left( \chi \right) + \frac{1}{w^{2}} M_{2}^{\rho }\left( \chi \right) + \frac{2}{w}M_{1}^{\rho }\left( \chi \right) \right] . \end{aligned}$$
(5.5)

Finally, taking supremum with respect to \(f\in B_{\varphi }\left( \mathbb {R}\right) \) with \(\left\| f \right\| _{\varphi }\le 1\) in (5.5) we have desired. \(\square \)

Next two theorem concerns some approximation properties of the operators \(G_{w}^{\chi , \rho }\) in weighted spaces of functions.

Theorem 7

Let \(\chi \in \psi \) be a \(\rho \)-kernel with \(M_2^\rho \left( \chi \right) <\infty \) and \(f\in C_{\varphi }\left( \mathbb {R}\right) \). Then,

$$\begin{aligned} \lim \limits _{w\rightarrow \infty } \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) = f\left( x\right) \end{aligned}$$
(5.6)

holds for every \(x\in \mathbb {R}\).

Proof

For all \(x\in \mathbb {R}\), \(k\in \mathbb {Z}\) and \(w >0,\) by a direct computation, we have the inequality

$$\begin{aligned}&\left| \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) -f\left( x\right) \right| \\ \le&\frac{\left| \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) \right| }{ \left( \varphi \circ \rho ^{-1} \right) \left( \frac{k}{w}\right) } \left| \left( \varphi \circ \rho ^{-1} \right) \left( \frac{k}{w}\right) -\varphi \left( x\right) \right| + \varphi \left( x\right) \left| \frac{\left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) }{\left( \varphi \circ \rho ^{-1} \right) \left( \frac{k}{w}\right) } - \frac{f\left( x\right) }{\varphi \left( x\right) } \right| . \end{aligned}$$

Then using the above inequality, we can write what follows:

$$\begin{aligned}{} & {} \left| \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right| \nonumber \\\le{} & {} \sum _{k\in \mathbb {Z}}\left| \chi \left( w\rho \left( x\right) -k\right) \right| \left[ \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) -f\left( x\right) \right] \nonumber \\\le{} & {} \sum _{k\in \mathbb {Z}} \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left\{ \frac{\left| \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) \right| }{\left( \varphi \circ \rho ^{-1} \right) \left( \frac{k}{w}\right) } \left| \left( \varphi \circ \rho ^{-1} \right) \left( \frac{k}{w}\right) -\varphi \left( x\right) \right| \right. \nonumber \\ {}{} & {} \left. +\varphi \left( x\right) \left| \frac{\left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) }{\left( \varphi \circ \rho ^{-1} \right) \left( \frac{k}{w}\right) } - \frac{f\left( x\right) }{\varphi \left( x\right) } \right| \right\} \nonumber \\:={} & {} I_{1} + I_{2}. \end{aligned}$$
(5.7)

Let us first estimate \(I_{1}\). Since \(f\in C_{\varphi }\left( \mathbb {R}\right) \), we have

$$\begin{aligned} I_{1}&\le \left\| f \right\| _{\varphi } \sum _{k\in \mathbb {Z}} \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left| \left( \frac{k}{w}\right) ^{2} - \rho ^{2}\left( x\right) \right| \\&\le \left\| f \right\| _{\varphi } \sum _{k\in \mathbb {Z}} \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left[ \left| \frac{k}{w} - \rho \left( x\right) \right| ^{2} + 2\left| \rho \left( x\right) \right| \left| \frac{k}{w}-\rho \left( x\right) \right| \right] \\&\le \frac{\left\| f \right\| _{\varphi }}{w^{2}}M_{2}^{\rho }\left( \chi \right) + \frac{2\left| \rho \left( x\right) \right| \left\| f \right\| _{\varphi }}{w}M_{1}^{\rho }\left( \chi \right) . \end{aligned}$$

Let us now consider \(I_{2}\). Let \(x\in \mathbb {R}\) and \(\varepsilon >0\) be fixed. Since f is continuous at x, \(f\circ \rho ^{-1}\) is continuous at \(\rho \left( x\right) \) and so \(\frac{f\circ \rho ^{-1}}{\varphi \circ \rho ^{-1}}\) is also continuous at \(\rho \left( x\right) \). So, there exists \(\delta >0\) such that \(\left| \frac{\left( f \circ \rho ^{-1}\right) \left( \frac{k}{w}\right) }{\left( \varphi \circ \rho ^{-1}\right) \left( \frac{k}{w}\right) } - \frac{\left( f\circ \rho ^{-1}\right) \left( \rho \left( x\right) \right) }{\left( \varphi \circ \rho ^{-1}\right) \left( \rho \left( x\right) \right) } \right| < \varepsilon \) whenever \(\left| \frac{k}{w}-\rho \left( x\right) \right| <\delta \). Then we can write

$$\begin{aligned} I_{2}&= \varphi \left( x\right) \sum _{\left| k-w\rho \left( x\right) \right| <w\delta } \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left| \frac{\left( f \circ \rho ^{-1}\right) \left( \frac{k}{w}\right) }{\left( \varphi \circ \rho ^{-1}\right) \left( \frac{k}{w}\right) } - \frac{\left( f\circ \rho ^{-1}\right) \left( \rho \left( x\right) \right) }{\left( \varphi \circ \rho ^{-1}\right) \left( \rho \left( x\right) \right) } \right| \\&+ \varphi \left( x\right) \sum _{\left| k-w\rho \left( x\right) \right| \ge w\delta } \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left| \frac{\left( f \circ \rho ^{-1}\right) \left( \frac{k}{w}\right) }{\left( \varphi \circ \rho ^{-1}\right) \left( \frac{k}{w}\right) } - \frac{\left( f\circ \rho ^{-1}\right) \left( \rho \left( x\right) \right) }{\left( \varphi \circ \rho ^{-1}\right) \left( \rho \left( x\right) \right) } \right| \\&:= J_{1} + J_{2}. \end{aligned}$$

It is easy to see that

$$\begin{aligned} J_{1} < \varepsilon \varphi \left( x\right) M_{0}^{\rho } \left( \chi \right) . \end{aligned}$$

For the case \(J_{2}\), by Remark 1 (i), we have for sufficiently large \(w>0\) that

$$\begin{aligned} J_{2} \le 2\left\| f \right\| _{\varphi }\varphi \left( x\right) \varepsilon . \end{aligned}$$

Finally, substituting the cases \(I_{1}\) and \(I_{2}\) in (5.7) we have

$$\begin{aligned} \begin{aligned}&\left| \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right| \\ \le&\frac{\left\| f \right\| _{\varphi }}{w^{2}}M_{2}^{\rho }\left( \chi \right) + \frac{2\left| \rho \left( x\right) \right| \left\| f \right\| _{\varphi }}{w}M_{1}^{\rho }\left( \chi \right) + \varepsilon \left( \varphi \left( x\right) M_{0}^{\rho } \left( \chi \right) + 2\left\| f \right\| _{\varphi }\varphi \left( x\right) \right) . \end{aligned} \end{aligned}$$
(5.8)

Taking the limit of both sides as \(w\rightarrow \infty \) we have (5.6). \(\square \)

Theorem 8

Let \(\chi \in \psi \) be a \(\rho \)-kernel with \(M_2^\rho \left( \chi \right) <\infty \) and \(\frac{f\circ \rho ^{-1}}{\varphi \circ \rho ^{-1}} \in U_{\varphi }\left( \mathbb {R}\right) \), then

$$\begin{aligned} \lim _{w\rightarrow \infty }\left\| G_{w}^{\chi , \rho } f-f\right\| _{\varphi }=0 \end{aligned}$$

holds.

Proof

For functions \(f\in U_{\varphi }\left( \mathbb {R}\right) \), let us follow the same steps with the proof of Theorem 7 and replace \(\delta \) with corresponding parameter of the uniform continuity of \(\frac{f\circ \rho ^{-1}}{\varphi \circ \rho ^{-1}}\). Also considering inequality (5.8) we have

$$\begin{aligned} \frac{\left| \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right| }{\varphi \left( x\right) } \le&\frac{\left\| f \right\| _{\varphi }}{\varphi \left( x\right) w^{2}}M_{2}^{\rho }\left( \chi \right) + \frac{2\left| \rho \left( x\right) \right| \left\| f \right\| _{\varphi }}{\varphi \left( x\right) w}M_{1}^{\rho }\left( \chi \right) \\&+ \varepsilon \left( M_{0}^{\rho } \left( \chi \right) + 2\left\| f \right\| _{\varphi }\right) \end{aligned}$$

and passing to supremum in the last inequality over \(x\in \mathbb {R}\), we have the desired result for \(w\rightarrow \infty \). \(\square \)

Now, we give the rate of convergence of the family of operators \(\left( G_{w}^{\chi ,\rho }\right) \) in terms of the weighted modulus of continuity given in (5.1).

Theorem 9

Let \(\chi \in \psi \) be a \(\rho \)-kernel with \(M_3^\rho \left( \chi \right) <\infty \). Then for \(f\circ \rho ^{-1}\in C_{\varphi }\left( \mathbb {R}\right) \), we get

$$\begin{aligned} \left| \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right| \le 9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2 \omega _\varphi \left( f\circ \rho ^{-1};w^{-1}\right) \left( M_{0}^{\rho }\left( \chi \right) + M_{3}^{\rho }\left( \chi \right) \right) . \end{aligned}$$

Proof

Using the definition of the operators \(G_{w}^{\chi ,\rho }\) and (5.4), we have

$$\begin{aligned}&\left| \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right| \\ \le&\sum _{k\in \mathbb {Z}} \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left[ \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) -f\left( x\right) \right] \\ \le&9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2 \omega _\varphi \left( f\circ \rho ^{-1};\delta \right) \sum _{k\in \mathbb {Z}} \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left( 1 + \frac{1}{\delta ^{3}}\left| \frac{k}{w}-\rho \left( x\right) \right| ^{3}\right) \\ \le&9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2 \omega _\varphi \left( f\circ \rho ^{-1};\delta \right) \left( M_{0}^{\rho }\left( \chi \right) + \frac{1}{\delta ^{3}w^{3}}M_{3}^{\rho }\left( \chi \right) \right) \end{aligned}$$

for \(f\circ \rho ^{-1} \in C_{\varphi }\left( \mathbb {R}\right) \) and \(\delta \le 1\). Choosing \(\delta = w^{-1}, w\ge 1\), we get

$$\begin{aligned} \left| \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right| \le 9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2 \omega _\varphi \left( f\circ \rho ^{-1};w^{-1}\right) \left( M_{0}^{\rho }\left( \chi \right) + M_{3}^{\rho }\left( \chi \right) \right) \end{aligned}$$

which is the desired result. \(\square \)

Corollary 2

Let \(\chi \in \psi \) be a \(\rho \)-kernel with \(M_3^\rho \left( \chi \right) <\infty \). Then, for \(f\circ \rho ^{-1}\in U_{\varphi }\left( \mathbb {R}\right) \), in view of (5.2), we get

$$\begin{aligned} \lim \limits _{w\rightarrow \infty }\left\| G_{w}^{\chi ,\rho }f-f\right\| _{\varphi } = 0. \end{aligned}$$

Remark 6

In Theorem 8, we stated the uniform convergence of \(G_{w}^{\chi ,\rho }\) for functions \(\frac{f\circ \rho ^{-1}}{\varphi \circ \rho ^{-1}}\in U_{\varphi }\left( \mathbb {R}\right) \). As a conclusion of Theorem 9, by using the property of weighted modulus of continuity, we obtained uniform convergence of \(G_{w}^{\chi ,\rho }\) for functions \(f\circ \rho ^{-1}\in U_{\varphi }\left( \mathbb {R}\right) \) in Corollary 2.

Here, we note that while the class of target functions in Theorem 8 is larger than the class of target functions considered in Corollary 2, the assumptions on the absolute moments imposed in Corollary 2 are stronger than the corresponding ones in Theorem 8.

As a final main result, we present quantitative form of the Voronovskaja-type formula in the weighted spaces of functions.

Theorem 10

Let \(\chi \in \psi \) be a \(\rho \)-kernel with \(M_4^\rho \left( \chi \right) <\infty \) and the first order \(\rho \)-algebraic moment of \(\chi \) is independent from x, i.e.,

$$\begin{aligned} m_{1}^\rho \left( \chi , x \right) =m_{1}^\rho \left( \chi \right) \in \mathbb {R}\backslash \{0\} \end{aligned}$$

for every \(x\in \mathbb {R}\). If \(f'\) and \(\rho '\) exists and \(\frac{f^{\prime }}{\rho ^{\prime }}\in C_{\varphi }\left( \mathbb {R}\right) \), then we have

$$\begin{aligned} \begin{aligned}&\left| w\left[ \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right] - \dfrac{f^{\prime }\left( x\right) }{\rho ^{\prime }\left( x\right) }m_{1}^{\rho }\left( \chi \right) \right| \\ \le&9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2\omega _\varphi \left( \frac{f^{\prime }}{\rho ^{\prime }};w^{-1}\right) \left[ M_{1}^{\rho }\left( \chi \right) +M_{4}^{\rho }\left( \chi \right) \right] \end{aligned} \end{aligned}$$
(5.9)

at any \(x\in \mathbb {R}\).

Proof

By the Taylor expansion of \(f\circ \rho ^{-1}\), we can write

$$\begin{aligned} \left( f\circ \rho ^{-1}\right) \left( \rho \left( u\right) \right) =&\left( f\circ \rho ^{-1}\right) \left( \rho \left( x\right) \right) + \left( f\circ \rho ^{-1}\right) ^{\prime }\\&\left( \rho \left( x\right) \right) \left( \rho \left( u\right) -\rho \left( x\right) \right) + R_{1}\left( f;u,x\right) , \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} R_{1}\left( f;u,x\right)&:= \left( \left( f\circ \rho ^{-1}\right) ^{\prime }\left( \rho \left( \xi \right) \right) - \left( f\circ \rho ^{-1}\right) ^{\prime }\left( \rho \left( x\right) \right) \right) \left( \rho \left( u\right) -\rho \left( x\right) \right) \\&= \left( \frac{f^{\prime }\left( \xi \right) }{\rho ^{\prime }\left( \xi \right) } - \frac{f^{\prime }\left( x\right) }{\rho ^{\prime }\left( x\right) }\right) \left( \rho \left( u\right) - \rho \left( x\right) \right) \end{aligned} \end{aligned}$$
(5.10)

and \(\xi \) is a number between u and x. Using the above Taylor formula in the definition of the operators \(G_{w}^{\chi ,\rho }\), we obtain

$$\begin{aligned} \left( G_{w}^{\chi ,\rho }f\right) \left( x\right)= & {} \sum _{k\in \mathbb {Z}}\chi \left( w\rho \left( x\right) -k\right) \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) \nonumber \\= & {} \sum _{k\in \mathbb {Z}} \chi \left( w\rho \left( x\right) -k\right) \left[ f\left( x\right) + \left( f\circ \rho ^{-1}\right) ^{\prime }\left( \rho \left( x\right) \right) \left( \frac{k}{w} - \rho \left( x\right) \right) \right] \nonumber \\{} & {} + R_{1}\left( f;\rho ^{-1}\left( \frac{k}{w}\right) ; x\right) \nonumber \\:= & {} I_{1} + I_{2}. \end{aligned}$$
(5.11)

It is clear that

$$\begin{aligned} I_{1} = f\left( x\right) + \dfrac{f^{\prime }\left( x\right) }{\rho ^{\prime }\left( x\right) }\dfrac{1}{w}m_{1}^{\rho }\left( \chi \right) . \end{aligned}$$

To estimate \(I_{2}\), if we use the inequality (5.4) and (5.10), we get

$$\begin{aligned} \left| I_{2}\right|&\le 9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2\omega _\varphi \left( \frac{f^{\prime }}{\rho ^{\prime }};\delta \right) \\&\quad \times \sum _{k\in \mathbb {Z}} \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left[ \left| \frac{k}{w}-\rho \left( x\right) \right| + \frac{1}{\delta ^{3}}\left| \frac{k}{w}-\rho \left( x\right) \right| ^{4}\right] \\&\le 9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2\omega _\varphi \left( \frac{f^{\prime }}{\rho ^{\prime }};\delta \right) \\&\quad \times \left[ \frac{1}{w}\sum _{k\in \mathbb {Z}} \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left| k-w\rho \left( x\right) \right| \right. \\&\quad \left. + \frac{1}{\delta ^{3}w^{4}}\sum _{k\in \mathbb {Z}} \left| \chi \left( w\rho \left( x\right) -k\right) \right| \left| k-w\rho \left( x\right) \right| ^{4}\right] . \end{aligned}$$

Now, choosing \(\delta =w^{-1}\), we have

$$\begin{aligned} \left| I_{2}\right| \le 9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2\omega _\varphi \left( \frac{f^{\prime }}{\rho ^{\prime }};w^{-1}\right) \left[ \frac{1}{w}M_{1}^{\rho }\left( \chi \right) +\frac{1}{w}M_{4}^{\rho }\left( \chi \right) \right] . \end{aligned}$$

Finally, substituting \(I_{1}\) and \(I_{2}\) in (5.11) we have

$$\begin{aligned}&\left| w\left[ \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right] - \dfrac{f^{\prime }\left( x\right) }{\rho ^{\prime }\left( x\right) }m_{1}^{\rho }\left( \chi \right) \right| \\&\qquad \le 9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2\omega _\varphi \left( \frac{f^{\prime }}{\rho ^{\prime }};w^{-1}\right) \left[ M_{1}^{\rho }\left( \chi \right) +M_{4}^{\rho }\left( \chi \right) \right] \end{aligned}$$

which is desired result. \(\square \)

Corollary 3

  1. 1.

    Let \(f\in C_{\varphi }\left( \mathbb {R}\right) \). If we choose \(\rho \left( x\right) =x\) in Theorem 10, we have the Voronovskaja theorem obtained in [3]:

    $$\begin{aligned} \lim \limits _{w\rightarrow \infty }w\left[ \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right] = f^{\prime }\left( x\right) m_{1}^{\rho }\left( \chi \right) ; \end{aligned}$$
  2. 2.

    Let \(\frac{f'}{\rho '}\in U_\varphi \left( \mathbb {R}\right) \). If we take limit of (5.9) as \(w\rightarrow \infty \), we have qualitative Voronovskaja-type theorem for \(G_{w}^{\chi , \rho }\), that is,

    $$\begin{aligned} \lim \limits _{w\rightarrow \infty }w\left[ \left( G_{w}^{\chi ,\rho }f\right) \left( x\right) -f\left( x\right) \right] = \dfrac{f^{\prime }\left( x\right) }{\rho ^{\prime }\left( x\right) }m_{1}^{\rho }\left( \chi \right) . \end{aligned}$$

6 Examples of Some \(\rho \)-Kernels

In this section, we present examples of some \(\rho \)-kernels satisfying the assumptions \((\chi 1), (\chi 2)\) and \((\chi 3)\). It is well-known that using Poisson-Summation formula given in [17], the assumption \((\chi 2)\) is equivalent to

$$\begin{aligned} \hat{\chi }\left( 2\pi k\right) =\left\{ \begin{array}{cc} 1, &{} k=0 \\ 0, &{} k\in \mathbb {Z}\backslash \left\{ 0\right\} \end{array}, \right. \end{aligned}$$
(6.1)

where \(\hat{\chi }\left( v\right) :=\int _{\mathbb {R}}\chi \left( y\right) e^{-ivy}dy,\ v\in \mathbb {R}\), is the Fourier transform of \(\chi \) (see [20, Lemma 4.2]).

6.1 Central B-Spline Kernel

For \(n\in \mathbb {N}\), central B-splines of order n are defined by

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

where \(\left( x\right) _+^{n-1}:=\max \left\{ x^{n-1},0\right\} \). The Fourier transform of \(B_n\) is

$$\begin{aligned} \hat{B}_n\left( v\right) =\left( \frac{\sin \left( \frac{v}{2}\right) }{\frac{v}{2}}\right) ^n, \ v\in \mathbb {R}. \end{aligned}$$

By considering the equality (6.1), we get

$$\begin{aligned} m_0^\rho \left( B_n,u\right) =1 \end{aligned}$$

for every \(u\in \mathbb {R}.\) Since central B-splines kernels have compact supports on \(\left[ \frac{-n}{2},\frac{n}{2}\right] \), all the absolute moments of arbitrary order \(\beta \) of \(B_n\) are finite.

As an example, we consider the 3-order B-spline:

$$\begin{aligned} B_3\left( x\right) =\left\{ \begin{array}{cc} \frac{3}{4}-x^2, &{} \left| x\right| \le \frac{1}{2}, \\ \frac{1}{2}\left( \frac{3}{2}-\left| x\right| \right) ^2, &{} \frac{1}{2}<\left| x\right| \le \frac{3}{2},\\ 0, &{} \left| x\right| >\frac{3}{2} \end{array} \right. \end{aligned}$$

for more details, see [20].

Corollary 4

For the modified generalized sampling series with central B-spline kernel we have

$$\begin{aligned} \left( G_{w}^{B_{n},\rho }f\right) \left( x\right) =\sum _{k\in \mathbb {Z}} \left( f\circ \rho ^{-1}\right) \left( \frac{k}{w}\right) B_{n}\left( w\rho \left( x\right) -k\right) \end{aligned}$$

and there holds:

i.:

for \(f\in CB\left( \mathbb {R}\right) \) (also for \(f\in C_{\varphi }\left( \mathbb {R}\right) \))

$$\begin{aligned} \lim \limits _{w\rightarrow \infty } \left( G_{w}^{B_{n},\rho }f\right) \left( x\right) = f\left( x\right) ; \end{aligned}$$
ii.:
$$\begin{aligned} \lim _{w\rightarrow \infty }\left\| G_{w}^{B_{n}, \rho } f-f\right\| _{\infty }=0 \end{aligned}$$

and

$$\begin{aligned} \lim _{w\rightarrow \infty }\left\| G_{w}^{B_{n}, \rho } f-f\right\| _{\varphi }=0 \end{aligned}$$

for \(f\circ \rho ^{-1}\in UC\left( \mathbb {R}\right) \) and \(\frac{f\circ \rho ^{-1}}{\varphi \circ \rho ^{-1}} \in U_{\varphi }\left( \mathbb {R}\right) \), respectively;

iii.:

if \(f\in CB\left( \mathbb {R}\right) \)

$$\begin{aligned} \left| \left( G_{w}^{B_{n},\rho } f\right) \left( x\right) -f\left( x\right) \right| \le \omega \left( f\circ \rho ^{-1},w^{-1}\right) \left( M_0^\rho \left( B_{n}\right) +M_1^\rho \left( B_{n}\right) \right) \end{aligned}$$

and if \(f\circ \rho ^{-1}\in C_{\varphi }\left( \mathbb {R}\right) \)

$$\begin{aligned}&\left| \left( G_{w}^{B_{n},\rho }f\right) \left( x\right) -f\left( x\right) \right| \\ \le&9\left( 1 + \left| \rho \left( x\right) \right| \right) ^2 \omega _\varphi \left( f\circ \rho ^{-1};w^{-1}\right) \left( M_{0}^{\rho }\left( B_{n}\right) + M_{3}^{\rho }\left( B_{n}\right) \right) ; \end{aligned}$$
iv.:

if \(f\in CB\left( \mathbb {R}\right) \) with \(f'\) and \(\rho '\) exists at \(x\in \mathbb {R}\) or \(\frac{f^{\prime }}{\rho ^{\prime }}\in U_{\varphi }\left( \mathbb {R}\right) \) with \(\rho '\) exists at \(x\in \mathbb {R}\) we have

$$\begin{aligned} \lim \limits _{w\rightarrow \infty }w\left[ \left( G_{w}^{B_{n},\rho }f\right) \left( x\right) -f\left( x\right) \right] = \dfrac{f^{\prime }\left( x\right) }{\rho ^{\prime }\left( x\right) }m_{1}^{\rho }\left( B_{n}\right) . \end{aligned}$$

6.2 \(\theta _\eta \)-Kernel

Let us consider the function \(\eta :\mathbb {R}\rightarrow \mathbb {R}\) given by

$$\begin{aligned} \eta \left( v\right) :=\left\{ \begin{array}{cc} 1, &{} v=0, \\ e^{-\frac{1}{e^{1/v^2}-e}}, &{}\left| v\right| <1,\ x\not =0,\\ 0, &{} \left| v\right| \ge 1. \end{array} \right. \end{aligned}$$

Then \(\theta \)-kernel is defined by

$$\begin{aligned} \theta _\eta \left( x\right) =\frac{1}{2\pi }\int _{-1}^{1}\eta \left( v\right) \cos \left( xv\right) dv. \end{aligned}$$

Again considering the equality (6.1), one can show that

$$\begin{aligned} m_0^\rho \left( \theta _\eta ,u\right) =1 \end{aligned}$$

for every \(u\in \mathbb {R}\). Since the \(\eta \) function has compact support on \(\left[ -1,1\right] \), \(\theta _\eta \) is a band-limited kernel. In addition, we have that \(\theta _\eta \left( x\right) =\mathcal {O}\left( \left| x\right| ^{-j}\right) \) as \(x\rightarrow \pm \infty \) for all \(j\in \mathbb {N}\cup \left\{ 0\right\} \). Then, from [11, Remark 3.2 (d)], we obtain the absolute moments of arbitrary order \(\beta \) of \(\theta _\eta \) are finite (for more details, see [28]).

Fig. 1
figure 1

Graph of \(\theta _\eta \) kernel

Full size image
Fig. 2
figure 2

Graph of function f and operators \(\left( G_{w}^{\chi }f\right) , \left( G_{w}^{\chi ,\rho _{1}}f\right) \) with \(w=10\) and B-spline kernel

Full size image
Fig. 3
figure 3

Graph of function g and operators \(\left( G_{w}^{\chi }g\right) , \left( G_{w}^{\chi ,\rho }g\right) \) with \(w=10\) and B-spline kernel

Full size image

For more kernel examples that are not given here, such as translates of central B-spline kernel, Fejer kernel, Bochner–Riesz kernel and Jackson kernel, we refer the readers to [11, 13, 18, 20].

7 Graphical Representations

Final section is devoted to give examples of graphical representations and numerical tables to compare the modified sampling operators and the classical sampling operators using the central B-spline kernel. These results can also be obtained by taking the other kernels which satisfy the assumptions of Theorem 1. According to these examples, we can see that newly constructed operators are better in approach than the old ones in some cases. Throughout the examples, we consider \(\rho _{1}:\mathbb {R}\rightarrow \mathbb {R}\) and \(\rho _{2}:\mathbb {R}\rightarrow \mathbb {R}\) functions given by

Table 1 Comparison of error of approximations of the classical generalized sampling series and the modified generalized sampling series by \(\rho _1\) and \(\rho _2\) for function f at some random values
Full size table
$$\begin{aligned} \rho _{1}\left( x\right) :=x^3+x \end{aligned}$$

and

$$\begin{aligned} \rho _2\left( x\right) := \dfrac{x^3+2x}{3}. \end{aligned}$$

It is easy to see that \(\rho _{1}\) and \(\rho _{2}\) satisfy the conditions \(\rho _1),\rho _2)\) and \(\rho _3)\).

Now let us consider the function \(f:\mathbb {R}\rightarrow \mathbb {R}, f\left( x\right) =\frac{x^2}{1+\left| x\right| ^3}\). Then we have the Fig. 2.

Secondly, we consider the function \(g:\mathbb {R}\rightarrow \mathbb {R},\ g\left( x\right) =\frac{1}{1+\left( x+2\right) ^2}\). Then we have the Fig. 3.

Finally, using \(\rho _{1}, \rho _{2}\) and f as target function we obtain some numerical results using central B-spline kernel of order 3 given in Table 1.