Abstract
In the present paper, we introduce a new family of sampling operators, so-called “modified sampling operators”, by taking a function \(\rho \) that satisfies the suitable conditions, and we study pointwise and uniform convergence of the family of newly introduced operators. We give the rate of convergence of the family of operators via classical modulus of continuity. We also obtain an asymptotic formula in the sense of Voronovskaja. Moreover, we investigate the approximation properties of modified sampling operators in weighted spaces of continuous functions characterized by \(\rho \) function. Finally, we present examples of some kernels that satisfy the appropriate assumptions. At the end, we present some graphical and numerical representations by comparing the modified sampling operators and the classical sampling operators.
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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
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
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
Remark 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]).
-
(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
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
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
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
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
Let us first consider \(S_1\). Since \(f\circ \rho ^{-1}\) is continuous at \(\rho \left( x_0\right) \), we get
Now we estimate \(S_2\). In view of Remark 1 (i) with \(\beta =0\), we have
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
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
The modulus of continuity satisfies the following properties:
For every \(f\in CB\left( \mathbb {R}\right) \) and any \(\lambda >0\),
and moreover, if \(f\in UC\left( \mathbb {R}\right) \), then
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
Proof
Using the definition of the operators and (3.2), we have by direct computation that
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
Proof
By the Taylor expansion of \(f\circ \rho ^{-1}\) at the point \(\rho \left( x\right) \in \mathbb {R}\), we have
where h is a bounded function such that
Now, using the definition of the operators (2.1) and the equality (4.1), we get
Indeed, it is easy to see that
Now, let us estimate \(I_{2}\). We write
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
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
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
at any point \(x\in \mathbb {R}\) for all \(w>w'\). Then,
Conversely, if (4.5) holds with strict inequalities at a given point \({x\in \mathbb {R}}\), then there exists \(w'>0\) such that \(w>w'\)
for \(w>w'\).
Proof
By the assumption (4.4), we have the inequality
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
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
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:
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
for each \(f \in C_{\varphi }\left( \mathbb {R}\right) \) and for every \(\delta >0\). We observe that
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
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 \(\delta >0\)
holds for all \(x, y \in \mathbb {R}\).
Remark 5
If we consider inequality (5.3), since
and
we get
If we combine two cases of \(\left| \rho \left( u\right) -\rho \left( x\right) \right| \) with respect to \(\delta \), it turns out that
Hence, choosing \(\delta \le 1\), we obtain
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:
Proof
For a fixed \(w>0\) and \(x\in \mathbb {R}\), using the definition of the operators \(G_{w}^{\chi ,\rho }\), we can write
which implies that
for every \(x\in \mathbb {R}\) and taking supremum over \(x\in \mathbb {R}\), we have
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,
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
Then using the above inequality, we can write what follows:
Let us first estimate \(I_{1}\). Since \(f\in C_{\varphi }\left( \mathbb {R}\right) \), we have
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
It is easy to see that
For the case \(J_{2}\), by Remark 1 (i), we have for sufficiently large \(w>0\) that
Finally, substituting the cases \(I_{1}\) and \(I_{2}\) in (5.7) we have
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
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
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
Proof
Using the definition of the operators \(G_{w}^{\chi ,\rho }\) and (5.4), we have
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
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
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.,
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
at any \(x\in \mathbb {R}\).
Proof
By the Taylor expansion of \(f\circ \rho ^{-1}\), we can write
where
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
It is clear that
To estimate \(I_{2}\), if we use the inequality (5.4) and (5.10), we get
Now, choosing \(\delta =w^{-1}\), we have
Finally, substituting \(I_{1}\) and \(I_{2}\) in (5.11) we have
which is desired result. \(\square \)
Corollary 3
-
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.
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
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
where \(\left( x\right) _+^{n-1}:=\max \left\{ x^{n-1},0\right\} \). The Fourier transform of \(B_n\) is
By considering the equality (6.1), we get
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:
for more details, see [20].
Corollary 4
For the modified generalized sampling series with central B-spline kernel we have
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
Then \(\theta \)-kernel is defined by
Again considering the equality (6.1), one can show that
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]).
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
and
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.
Data Availability
All data generated or analyzed during this study are included in this published article. All authors reviewed the manuscript.
Notes
This modulus of continuity is originally given for \(x,t >0\), but we can generalize it to \(x,t\in \mathbb {R}\) without any difference.
References
Acar, T.: Asymptotic formulas for generalized Szász–Mirakyan operators. Appl. Math. Comput. 263, 233–239 (2015)
Acar, T., Aral, A., Raşa, I.: Positive linear operators preserving \(\tau \) and \(\tau ^2\). Constr. Math. Anal. 2(3), 98–102 (2019)
Acar, T., Alagöz, O., Aral, A., Costarelli, D., Turgay, M., Vinti, G.: Convergence of generalized sampling series in weighted spaces. Demonstr. Math. 55, 153–162 (2022)
Acar, T., Eke, A., Kursun, S.: Bivariate generalized Kantorovich-type exponential sampling series, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 118, 35 (2024)
Acar, T., Kursun, S., Turgay, M.: Multidimensional Kantorovich modifications of exponential sampling series. Quaest. Math. 46(1), 57–72 (2023)
Acar, T., Kursun, S.: Pointwise convergence of generalized Kantorovich exponential sampling series. Dolomites Res. Notes Approx. 16, 1–10 (2023)
Acar, T., Ulusoy, G.: Approximation by modified Szász–Durrmeyer operators. Period. Math. Hung. 72, 64–75 (2016)
Alagoz, O., Turgay, M., Acar, T., Parlak, M.: Approximation by sampling durrmeyer operators in weighted space of functions. Numer. Funct. Anal. Optim. 43(10), 1223–1239 (2022)
Angeloni, L., Cetin, N., Costarelli, D., Sambucini, A.R., Vinti, G.: Multivariate sampling Kantorovich operators: quantitative estimates in Orlicz spaces. Constr. Math. Anal. 4(2), 229–241 (2021)
Aral, A., Acar, T., Kursun, S.: Generalized Kantorovich forms of exponential sampling series. Anal. Math. Phys. 12, 50 (2022)
Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampl. Theory Signal Image Process. 6(1), 29–52 (2007)
Bardaro, C., Faina, L., Mantellini, I.: A generalization of the exponential sampling series and its approximation properties. Math. Slovaca 67(6), 1481–1496 (2017)
Bardaro, C., Mantellini, I.: Asymptotic formulae for linear combinations of generalized sampling type operators. Zeitschrift für Anal. und ihre Anwendung 32, 279–298 (2013)
Bardaro, C., Mantellini, I.: Asymptotic expansion of generalized Durrmeyer sampling type series. Jaen J. Approx. 6(2), 143–165 (2014)
Bernstein, S.N.: Démonstration du théorème de Weierstrass, fondée sur le calcul des probabilités., Math. Charkow (2), 13(1-2), in French (1912)
Butzer, P.L., Engels, W., Ries, S., Stens, R.L.: The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines. SIAM J. Appl. Math. 46(2), 299–323 (1986)
Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation I. Academic Press, New York-London (1971)
Butzer, P.L., Schmeisser, G., Stens, R.L.: Basic relations valid for the Bernstein space \(B^p\) and their extensions to functions from larger spaces with error estimates in term of their distances from \(B^p\). J. Fourier Anal. Appl. 19, 333–375 (2013)
Butzer, P.L., Splettstosser, W.: A sampling theorem for duration-limited functions with error estimates. Inf. Control. 34(1), 55–65 (1977)
Butzer, P.L., Splettstosser, W., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Deutsch. Math.-Verein. 90, 1–70 (1988)
Butzer, P.L., Stens, R. L.: Linear prediction by samples from the past. In: Advanced topics in Shannon sampling and interpolation theory, New York, Springer, 157–183 (1993)
Butzer, P.L., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Dtsch. Math. Ver. 90(1), 1–70 (1998)
Cárdenas-Morales, D., Garrancho, P., Raşa, I.: Bernstein-type operators which preserve polynomials. Comput. Math. with Appl. 62, 158–163 (2011)
Costarelli, D., Piconi, M., Vinti, G.: On the convergence properties of sampling Durrmeyer-type operators in Orlicz spaces. Mathematische Nachrichten 296(2), 588–609 (2023)
Costarelli, D., Vinti, G.: Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. J. Integr. Equ. Appl. 26(4), 455–481 (2014)
Costarelli, D., Vinti, G.: a quantitative estimate for the sampling Kantorovich series in terms of the modulus of continuity in Orlicz spaces. Constr. Math. Anal. 2(1), 8–14 (2019)
DeVore, R.A., Lorentz, G.G.: Constructive approximation, vol. 303. Springer, Berlin (1993)
Draganov, B.R.: A fast converging sampling operator. Constr. Math. Anal. 5(4), 190–201 (2022)
Erencin, A., Raşa, I.: Voronovskaya type theorems in weighted spaces. Numer. Funct. Anal. Optim. 37(12), 1517–1528 (2016)
Holhos, A.: Quantitative estimates for positive linear operators in weighted space. Gen. Math. 16(4), 99–110 (2008)
Ilarslan, H.G.I., Başcanbaz-Tunca, G.: Convergence in variation for Bernstein-type operators. Mediterr. J. Math. 13(5), 2577–2592 (2016)
King, J.P.: Positive linear operators which preserve \(x^2\). Acta. Math. Hungar. 99, 203–208 (2003)
Kotel’nikov, V.A.: On the carrying capacity of “ether” and wire in electrocommunications, Material for the First All-Union Conference on the Questions of Communications, Moscow (1933)
Acar T., Kursun S., Acar O.: Approximation properties of exponential sampling series in logarithmic weighted spaces. B. Iran Math. Soc. 50, 36 (2024).
Kursun, S., Aral, A., Acar, T.: Approximation results for Hadamard-type exponential sampling Kantorovich series. Mediterr. J. Math. 20, 263 (2023)
Kursun, S., Aral, A., Acar, T.: Riemann-Liouville fractional integral type exponential sampling Kantorovich series, Exp. Syst. Appl., 238(F), 122350 (2024)
Kursun, S., Turgay, M., Alagoz, O., Acar, T.: Approximation Properties of Multivariate Exponential Sampling Series. Carpathian Math. Publ. 13(3), 666–675 (2021)
Özer, D., Turgay, M., Acar, T.: Approximation properties of bivariate sampling Durrmeyer series in weighted spaces of functions. Adv. Stud. Euro-Tbil. Math. J. 16(Supp. 3), 89–107 (2023)
Riesz, S., Stens, R.L.: Approximation by generalized sampling series, Bl. Sendov et al., Publ. House Bulgarian Acad. Sci (Sofia), 746–756 (1984)
Shannon, C.E.: Communications in the presence of noise. Proc. IRE. 37, 10–21 (1949)
Whittaker, E.T.: On the functions, which are represented by expansions of the interpolation theory. Proc. R. Soc. Edinb. 35, 181–194 (1915)
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This study was supported by Scientific and Technological Research Council of Turkey (TUBITAK) under the Grant Number 123F123. The authors thank to TUBITAK for their supports.
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Turgay, M., Acar, T. Approximation by Modified Generalized Sampling Series. Mediterr. J. Math. 21, 107 (2024). https://doi.org/10.1007/s00009-024-02653-w
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DOI: https://doi.org/10.1007/s00009-024-02653-w