![Graph of θη\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _\eta $$\end{document} kernel](https://www.researchgate.net/publication/379724770/figure/fig1/AS:11431281247272985@1716688130003/Graph-of-thedocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
![Graph of function f and operators Gwχf,Gwχ,ρ1f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( G_{w}^{\chi }f\right) , \left( G_{w}^{\chi ,\rho _{1}}f\right) $$\end{document} with w=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w=10$$\end{document} and B-spline kernel](https://www.researchgate.net/publication/379724770/figure/fig2/AS:11431281247151725@1716688130411/Graph-of-function-f-and-operators-Gwchf-Gwch-r1fdocumentclass12ptminimal_Q320.jpg)
![Graph of function g and operators Gwχg,Gwχ,ρg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( G_{w}^{\chi }g\right) , \left( G_{w}^{\chi ,\rho }g\right) $$\end{document} with w=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w=10$$\end{document} and B-spline kernel](https://www.researchgate.net/publication/379724770/figure/fig3/AS:11431281247272986@1716688130647/Graph-of-function-g-and-operators-Gwchg-Gwch-rgdocumentclass12ptminimal_Q320.jpg)
Access to this full-text is provided by Springer Nature.
Content available from Mediterranean Journal of Mathematics
This content is subject to copyright. Terms and conditions apply.
Mediterr. J. Math. (2024) 21:107
https://doi.org/10.1007/s00009-024-02653-w
1660-5446/24/030001-22
published online April 30, 2024
c
The Author(s) 2024
Approximation by Modified Generalized
Sampling Series
Metin Turgay and Tuncer Acar
Abstract. In the present paper, we introduce a new family of sampling
operators, so-called “modified sampling operators”, by taking a func-
tion ρthat 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 mod-
ulus of continuity. We also obtain an asymptotic formula in the sense
of Voronovskaja. Moreover, we investigate the approximation proper-
ties of modified sampling operators in weighted spaces of continuous
functions characterized by ρfunction. 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.
Mathematics Subject Classification. 41A25, 41A35, 94A20, 41A81.
Keywords. Generalized sampling series, rate of convergence, modulus of
continuity, weighted approximation.
1. Introduction
Bernstein polynomials are useful tool to prove the well-known Weierstrass ap-
proximation theorem for the space of continuous functions on [0,1] or more
generally on [a, b]⊂R(see [15]). In [32], King constructed and studied a gen-
eralization of the classical Bernstein operators using a sequence of continuous
functions defined on [0,1], n∈Nto obtain a better approximation. In [23],
the authors introduced a new type of operators Bnin the form
Bρ
n(f;x):=
n
k=0 f◦ρ−1k
nn
kρk(x)(1−ρ(x))n−k,x∈[0,1] ,n ∈N,
using a special function ρ:[0,1] →Rthat satisfies suitable assumptions. In
the same paper, the authors obtained that a new family of operators gives a
better approach than the operators Bnin certain cases. Similar constructions
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 2 of 22 M. Turgay, T. Acar MJOM
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
(Gχ
wf)(x):=
k∈Z
fk
wχ(wx −k),x∈R,w > 0,(1.1)
where χ:R→Ris called a kernel function satisfying certain assumptions of
approximate identities and f:R→Ris a bounded, continuous function on R.
The generalized sampling operators given in (1.1), were considered as an ap-
proximate version of the classical Whittaker–Kotel’nikov–Shannon sampling
theorem (see [33,40,41]). In recent years, numerous studies have been pub-
lished on sampling type operators. We can refer the readers to [19,21,28,39]
for generalized sampling operators, [9,11,26] for sampling Kantorovich oper-
ators, [14,24] for sampling Durrmeyer operators, [5,6,12,37] for exponential
sampling type operators and [3,4,8,10,34–36,38] 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 ρfunction, 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. 6and 7, we give some examples of the kernels satisfying
suitable assumptions and by considering a special ρfunction, we present some
graphical and numerical representations to compare the modified generalized
sampling operators Gχ,ρ
wand the classical generalized sampling operators Gχ
w.
2. Basic Notations and Preliminaries
By N,Zand R, we shall denote the sets of all positive integers, integers and
real numbers, respectively.
By C(R), we will denote the space of all continuous (not necessarily
bounded) functions defined on Rand by CB (R) the space of all bounded
functions f∈C(R) endowed with the norm f:= supx∈R|f(x)|.Moreover,
by UC (R), we denote the subspaces of CB (R) comprising all uniformly
continuous functions.
Let ρ:R→Rbe a strictly increasing function that satisfies the follow-
ing conditions:
(ρ1)ρ∈C(R);
(ρ2)ρ(0) = 0, lim
x±∞ ρ(x)=±∞.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 3 of 22 107
Definition 1. Throughout the paper, a function χ:R→Ris said to be
a kernel associated with ρ(or simply ρ-kernel) if it satisfies the following
assumptions:
(χ1) χ∈C(R);
(χ2) for every u∈R, discrete ρ-algebraic moment of order 0 of χis 1, that is
mρ
0(χ, u)=
k∈Z
χ(ρ(u)−k)=1;
(χ3) for any β≥0, absolute moment of order βassociated with ρof χ(or
simpliy ρ-absolute moment) is finite, that is
Mρ
β(χ)=sup
u∈R
k∈Z|χ(ρ(u)−k)||k−ρ(u)|β<∞.
By ψ, we will denote the class of all functions satisfying the assumptions
(χ1),(χ2) and (χ3).
For any function χ:R→R, a discrete algebraic moment of order
j∈N∪{0}associated with ρof χ(or simply ρ-algebraic moment) is defined
by
mρ
j(χ, u):=
k∈Z
χ(ρ(u)−k)(k−ρ(u))j,u∈R.
Mρ
β(χ):=sup
u∈R
k∈Z|χ(ρ(u)−k)||k−ρ(u)|β,u∈R.
Remark 1.(i) Let χbe a function satisfying (χ1) and (χ3), there holds:
lim
w→∞
|k−wρ(x)|≥wδ |χ(wρ (x)−k)|=0
uniformly with respect to x∈R(see [11]).
(ii) For η, γ > 0 with η<γ,M
ρ
γ(χ)<∞implies Mρ
η(χ)<∞. When χ
has compact support, we immediately have that Mρ
γ(χ)<∞for every
γ>0(see[25]).
Now, we introduce a new family of sampling type operators, so-called
modified generalized sampling operators, by
(Gχ,ρ
wf)(x)=Gχ
wf◦ρ−1(ρ(x))
:=
k∈Zf◦ρ−1k
wχ(wρ (x)−k),x∈R,w > 0 (2.1)
for χ∈ψ.
Remark 2.The operator (2.1) is well-defined if, for example, fis bounded.
Indeed, if |f(x)|≤Lfor every x∈R, then f◦ρ−1is also a bounded function.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 4 of 22 M. Turgay, T. Acar MJOM
Then
|(Gχ,ρ
wf)(x)|≤
k∈Zf◦ρ−1k
w|χ(wρ (x)−k)|
≤L
k∈Z|χ(wρ (x)−k)|
≤LMρ
0(χ)<∞.
Remark 3.In the special case of ρ(x)=x(it is clear that (ρ1) and (ρ2)
are satisfied), the operators (2.1) reduce to the classical generalized sampling
series
(Gχ
wf)(x)=
k∈Z
fk
wχ(wx −k).
3. Approximation Results for Gχ,ρ
w
In this section, we present some approximation results for the family of oper-
ators (Gχ,ρ
w) including pointwise convergence, uniform convergence, and rate
of convergence.
Theorem 1. Let χ∈ψbe a ρ-kernel. If f:R→Ris a bounded function,
then
lim
w→∞ (Gχ,ρ
wf)(x0)=f(x0) (3.1)
holds at each continuity point x0∈Rof f.
Proof. Assume that x0be a continuity point of f.So,f◦ρ−1◦ρis also con-
tinuous at the point x0and that means f◦ρ−1is continuous at ρ(x0). Hence,
for every ε>0, there exists δ>0 such that f◦ρ−1k
w−f◦ρ−1(ρ(x0))<
εwhenever k
w−ρ(x0)<δ. Thus, we can write
|(Gχ,ρ
wf)(x0)−f(x0)|
≤
k∈Zf◦ρ−1k
w−f◦ρ−1(ρ(x0))|χ(wρ (x0)−k)|
=⎛
⎝
|k−wρ(x0)|<wδ
+
|k−wρ(x0)|≥wδ⎞
⎠f◦ρ−1k
w
−f◦ρ−1(ρ(x0))|χ(wρ (x0)−k)|
=: S1+S2.
LetusfirstconsiderS1. Since f◦ρ−1is continuous at ρ(x0), we get
S1≤εMρ
0(χ).
Now we estimate S2. In view of Remark 1(i) with β=0,wehave
S2≤2
f◦ρ−1
∞
|k−wρ(x0)|≥wδ |χ(wρ (x0)−k)|
<2
f◦ρ−1
∞ε
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 5 of 22 107
for sufficiently large w. Combining S1and S2and taking limit as w→∞we
conclude (3.1).
Theorem 2. Let χ∈ψbe a ρ-kernel. If f◦ρ−1∈UC (R), then
lim
w→∞ Gχ,ρ
wf−f∞=0
holds.
Proof. The proof follows the same argument as Theorem 1, taking into ac-
count that if f◦ρ−1∈UC (R), then we can choose δ>0 independent of x
such that for |k−wρ (x)|<wδ, one has f◦ρ−1k
w−f◦ρ−1(ρ(x))<
εuniformly with respect to x∈R.
Remark 4.We can not change the assumption f◦ρ−1∈UC (R)tof∈
UC (R) in Theorem 2, since uniform continuity of fdoes not guarantee uni-
form continuity of f◦ρ−1. For example, consider f:R→R,f(x)=xand
ρ(x)= 3
√x.
Now, we will give a quantitative estimate for functions f∈CB (R)via
the classical modulus of continuity. First, let us remind the definition of the
modulus of continuity. For functions f∈CB (R)andδ>0, the modulus of
continuity is defined by
ω(f,δ)= sup
t,x∈R
|t−x|<δ
|f(t)−f(x)|.
The modulus of continuity satisfies the following properties:
For eve r y f∈CB (R) and any λ>0,
ω(f,λδ)≤(1 + λ)ω(f,δ) (3.2)
and moreover, if f∈UC (R), then
lim
δ→0ω(f,δ) = 0 (3.3)
holds (see [27]).
Theorem 3. Let f∈CB (R).Ifχ∈ψbe a ρ-kernel with Mρ
1(χ)<∞, then
we have
|(Gχ,ρ
wf)(x)−f(x)|≤ωf◦ρ−1,w
−1(Mρ
0(χ)+Mρ
1(χ)) .(3.4)
Proof. Using the definition of the operators and (3.2), we have by direct
computation that
|(Gχ,ρ
wf)(x)−f(x)|≤
k∈Z
ωf◦ρ−1,
k
w−ρ(x)|χ(wρ (x)−k)|
≤
k∈Z1+|k−wρ (x)|
wδ ωf◦ρ−1,δ
|χ(wρ (x)−k)|
and choosing δ=w−1, we get the desired result.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 6 of 22 M. Turgay, T. Acar MJOM
4. Voronovska ja-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
χand ρto state and prove the Voronovskaja-type theorem:
(ρ3)Letρbe a continuously differentiable function
and
(χ4) Mρ
1(χ)<∞and
lim
w→∞
|k−wρ(x)|≥wδ |χ(wρ (x)−k)||k−wρ (x)|=0
holds uniformly with respect to x∈R.
There are many kernels that satisfy the assumption (χ4); for instance, Trans-
lates of B-splines, Bochner–Riesz kernel, generalized Jackson kernel, for de-
tails, see [13].
Theorem 4. Let f∈CB (R). Suppose that fand ρexistatanyx∈Rand
mρ
1(χ, x):=mρ
1(χ)=0is independent of x.Ifχ∈ψbe a ρ-kernel such that
(χ4) is satisfied, then we have
lim
w→∞ w[(Gχ,ρ
wf)(x)−f(x)] = f(x)
ρ(x)mρ
1(χ)+ow−1.
Proof. By the Taylor expansion of f◦ρ−1at the point ρ(x)∈R,wehave
f◦ρ−1k
w=f(x)+f◦ρ−1(ρ(x)) k
w−ρ(x)+hk
wk
w−ρ(x),
(4.1)
where his a bounded function such that
lim
k
w→ρ(x)
hk
w=0.(4.2)
Now, using the definition of the operators (2.1) and the equality (4.1), we get
(Gχ,ρ
wf)(x)−f(x)
=
k∈Z
χ(wρ (x)−k)f(x)
ρ(x)k
w−ρ(x)+hk
wk
w−ρ(x)
=
k∈Z
χ(wρ (x)−k)f(x)
ρ(x)k
w−ρ(x)
+
k∈Z
χ(wρ (x)−k)hk
wk
w−ρ(x)
=: I1+I2.
Indeed, it is easy to see that
I1=f(x)
ρ(x)
1
wmρ
1(χ).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 7 of 22 107
Now, let us estimate I2. We write
|I2|=⎛
⎝
|k−wρ(x)|<wδ
+
|k−wρ(x)|≥wδ⎞
⎠|χ(wρ (x)−k)|
hk
w
k
w−ρ(x)
:= I2,1+I2,2.
Using (4.2), we have that I2,1≤ε
wMρ
1(χ). Moreover, by using boundedness
of h, we have that I2,2≤h∞
wεfor sufficiently large wby Remark 1(i) with
β= 1. Hence, we conclude that
w[(Gχ,ρ
wf)(x)−f(x)] = f(x)
ρ(x)mρ
1(χ)+ ε
w(Mρ
1(χ)+h∞)
and the assertion follows as w→∞.
Using the similar methods applied in the proof of Theorem 4, the fol-
lowing Corollary can be proved:
Corollary 1. Let f∈CB (R). Suppose that f and ρ exist at any x∈R
and mρ
2(χ, x):=mρ
2(χ)=0is independent of x. Suppose also that mρ
1(χ)
is independent of xand mρ
1(χ)=0.Ifχ∈ψbe a ρ-kernel such that (χ4) is
satisfied, then we have
lim
w→∞ w2[(Gχ,ρ
wf)(x)−f(x)] = f (x)
[ρ(x)]2−f(x)ρ (x)
[ρ(x)]3mρ
2(χ).(4.3)
Theorem 5. Let f∈CB (R). Suppose that f,ρ
exists at any x∈R,m
ρ
1(χ)=
0,m
ρ
2(χ, x):=mρ
2(χ)>0is independent of x,χ∈ψis a ρ-kernel such that
(χ4) is satisfied. Assume that there exists w>0such that
f(x)≤(Gχ,ρ
wf)(x)≤(Gχ
wf)(x) (4.4)
at any point x∈Rfor all w>w
.Then,
f (x)≥ρ (x)
ρ(x)f(x)≥1−[ρ(x)]2f (x),x∈R.(4.5)
Conversely, if (4.5)holds with strict inequalities at a given point x∈R,
then there exists w>0such that w>w
f(x)<(Gχ,ρ
wf)(x)<(Gχ
wf)(x) (4.6)
for w>w
.
Proof. By the assumption (4.4), we have the inequality
0≤w2[(Gχ,ρ
wf)(x)−f(x)] ≤w2[(Gχ
wf)(x)−f(x)]
at any point x∈Rfor all w>w
. Then, using (4.3) (recall the classical
Voronovskaja theorem for Gχ
wby the fact that ρ(x)=xin (4.3)) we have
0≤f (x)
[ρ(x)]2−f(x)ρ (x)
[ρ(x)]3≤f (x)
which yields (4.5).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 8 of 22 M. Turgay, T. Acar MJOM
Conversely, if (4.5) holds with strict inequalities at any x∈R, then
multiplying each terms of inequality (4.5)by 1
[ρ(x)]2and subtract f (x)
[ρ(x)]2
from each terms, respectively, we have
0<f (x)
[ρ(x)]2−f(x)ρ (x)
[ρ(x)]3<f
(x)
and using again (4.3), desired result is obtained.
Example 1.Let us consider a function f:R→Rgiven by f(x)=x3
3and
ρ(x)=x3+x. Under these considerations inequality (4.5) holds for strict
inequalities for all x∈R\{0}. So we can say, theoretically, that modified
generalized sampling series gives a better approach than classical one for all
x∈R\{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 ϕ:R→R,ϕ(x)=1+ρ2(x), we shall
consider the following class of functions:
Bϕ(R)=f:R→R|for every x∈R,|f(x)|
ϕ(x)≤Mf,
Cϕ(R)=C(R)∩Bϕ(R),
Uϕ(R)=f∈Cϕ(R)||f(x)|
ϕ(x)is uniformly continuous on R,
where Mfis a constant depending only on fand the above spaces are normed
linear spaces with the norm fϕ=sup
x∈R
|f(x)|
ϕ(x). The weighted modulus of
continuity defined in [30]1is given by
ωϕ(f;δ)= sup
x,t∈R
|ρ(t)−ρ(x)|≤δ
|f(t)−f(x)|
ϕ(t)+ϕ(x)(5.1)
for each f∈Cϕ(R)andforeveryδ>0. We observe that
ωϕ(f;0) = 0
for every f∈Cϕ(R) and the function ωϕ(f;δ) is nonnegative and nonde-
creasing with respect to δfor f∈Cϕ(R) and also
lim
δ→0ωϕ(f;δ) = 0 (5.2)
for every f∈Uϕ(R) (for more details, see [30]). We recall the following
auxiliary lemma to obtain an estimate for |f(u)−f(x)|.
1This modulus of continuity is originally given for x, t > 0, but we can generalize it to
x, t ∈Rwithout any difference.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 9 of 22 107
Lemma 1. ([30]) For every f∈Cϕ(R)and δ>0
|f(u)−f(x)|≤(ϕ(u)+ϕ(x)) 2+|ρ(u)−ρ(x)|
δωϕ(f,δ) (5.3)
holds for all x, y ∈R.
Remark 5.If we consider inequality (5.3), since
ϕ(u)+ϕ(x)≤δ2+2ρ2(x)+2|ρ(x)|δwhenever |ρ(u)−ρ(x)|≤δ
and
ϕ(u)+ϕ(x)≤δ2+2ρ2(x)+2|ρ(x)|δ|ρ(u)−ρ(x)|
δ2
whenever |ρ(u)−ρ(x)|>δ,
we get
|f(u)−f(x)|
≤
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
δ2+2ρ2(x)+2|ρ(x)|δ+2
2+|ρ(u)−ρ(x)|
δωϕ(f;δ),|ρ(u)−ρ(x)|≤δ
δ2+2ρ2(x)+2|ρ(x)|δ+2
|ρ(u)−ρ(x)|
δ2
2+|ρ(u)−ρ(x)|
δωϕ(f;δ),|ρ(u)−ρ(x)|>δ
≤⎧
⎨
⎩
3δ2+2ρ2(x)+2|ρ(x)|δ+2
ωϕ(f;δ),|ρ(u)−ρ(x)|≤δ
3δ2+2ρ2(x)+2|ρ(x)|δ+2
|ρ(u)−ρ(x)|3
δ3ωϕ(f;δ),|ρ(u)−ρ(x)|>δ
.
If we combine two cases of |ρ(u)−ρ(x)|with respect to δ, it turns out that
|f(u)−f(x)|≤3δ2+2ρ2(x)+2|ρ(x)|δ+2
ωϕ(f;δ)1+|ρ(u)−ρ(x)|3
δ3.
Hence, choosing δ≤1, we obtain
|f(u)−f(x)|≤9(1+ |ρ(x)|)2ωϕ(f;δ)1+ |ρ(u)−ρ(x)|3
δ3.(5.4)
As a first main result of this section, we present the well-definiteness of
the family of operators (Gχ,ρ
w) in weighted spaces of functions.
Theorem 6. Let χ∈ψbe a ρ-kernel with Mρ
2(χ)<∞.Then,forafixed
w>0, the operator Gχ,ρ
wis a linear operator from Bϕ(R)to Bϕ(R)and its
operator norm turns out to be:
Gχ,ρ
wBϕ→Bϕ≤Mρ
0(χ)+ 1
w2Mρ
2(χ)+ 2
wMρ
1(χ).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 10 of 22 M. Turgay, T. Acar MJOM
Proof. For a fixed w>0andx∈R, using the definition of the operators
Gχ,ρ
w, we can write
|(Gχ,ρ
wf)(x)|≤
k∈Z|χ(wρ (x)−k)|fρ−1k
w
1+ρ2ρ−1k
w1+ρ2ρ−1k
w
≤fϕ
k∈Z|χ(wρ (x)−k)|1+k
w2
=fϕ
k∈Z|χ(wρ (x)−k)|1+k
w−ρ(x)2
+2ρ(x)k
w−ρ(x)+ρ2(x)
≤fϕ1+ρ2(x)
k∈Z|χ(wρ (x)−k)|1+ 1
w2(k−wρ (x))2
+2
w|k−wρ (x)|
≤fϕ1+ρ2(x)Mρ
0(χ)+ 1
w2Mρ
2(χ)+ 2
wMρ
1(χ)
which implies that
|(Gχ,ρ
wf)(x)|
1+ρ2(x)≤fϕMρ
0(χ)+ 1
w2Mρ
2(χ)+ 2
wMρ
1(χ)
for every x∈Rand taking supremum over x∈R,wehave
Gχ,ρ
wfϕ≤fϕMρ
0(χ)+ 1
w2Mρ
2(χ)+ 2
wMρ
1(χ).(5.5)
Finally, taking supremum with respect to f∈Bϕ(R) with fϕ≤1in(5.5)
we have desired.
Next two theorem concerns some approximation properties of the oper-
ators Gχ,ρ
win weighted spaces of functions.
Theorem 7. Let χ∈ψbe a ρ-kernel with Mρ
2(χ)<∞and f∈Cϕ(R).
Then,
lim
w→∞ (Gχ,ρ
wf)(x)=f(x) (5.6)
holds for every x∈R.
Proof. For all x∈R,k∈Zand w>0,by a direct computation, we have the
inequality
f◦ρ−1k
w−f(x)
≤f◦ρ−1k
w
(ϕ◦ρ−1)k
wϕ◦ρ−1k
w−ϕ(x)
+ϕ(x)f◦ρ−1k
w
(ϕ◦ρ−1)k
w−f(x)
ϕ(x)
.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 11 of 22 107
Then using the above inequality, we can write what follows:
|(Gχ,ρ
wf)(x)−f(x)|
≤
k∈Z|χ(wρ (x)−k)|f◦ρ−1k
w−f(x)
≤
k∈Z|χ(wρ (x)−k)|f◦ρ−1k
w
(ϕ◦ρ−1)k
wϕ◦ρ−1k
w−ϕ(x)
+ϕ(x)f◦ρ−1k
w
(ϕ◦ρ−1)k
w−f(x)
ϕ(x)
:= I1+I2.(5.7)
Let us first estimate I1. Since f∈Cϕ(R), we have
I1≤fϕ
k∈Z|χ(wρ (x)−k)|k
w2
−ρ2(x)
≤fϕ
k∈Z|χ(wρ (x)−k)|
k
w−ρ(x)
2
+2|ρ(x)|
k
w−ρ(x)
≤fϕ
w2Mρ
2(χ)+2|ρ(x)|fϕ
wMρ
1(χ).
Let us now consider I2.Letx∈Rand ε>0 be fixed. Since fis continuous
at x,f◦ρ−1is continuous at ρ(x)andso f◦ρ−1
ϕ◦ρ−1is also continuous at ρ(x).
So, there exists δ>0 such that
(f◦ρ−1)( k
w)
(ϕ◦ρ−1)(k
w)−(f◦ρ−1)(ρ(x))
(ϕ◦ρ−1)(ρ(x))
<εwhenever
k
w−ρ(x)<δ. Then we can write
I2=ϕ(x)
|k−wρ(x)|<wδ |χ(wρ (x)−k)|f◦ρ−1k
w
(ϕ◦ρ−1)k
w−f◦ρ−1(ρ(x))
(ϕ◦ρ−1)(ρ(x))
+ϕ(x)
|k−wρ(x)|≥wδ |χ(wρ (x)−k)|f◦ρ−1k
w
(ϕ◦ρ−1)k
w−f◦ρ−1(ρ(x))
(ϕ◦ρ−1)(ρ(x))
:= J1+J2.
It is easy to see that
J1<εϕ(x)Mρ
0(χ).
For the case J2, by Remark 1(i), we have for sufficiently large w>0 that
J2≤2fϕϕ(x)ε.
Finally, substituting the cases I1and I2in (5.7)wehave
|(Gχ,ρ
wf)(x)−f(x)|
≤fϕ
w2Mρ
2(χ)+2|ρ(x)|fϕ
wMρ
1(χ)+εϕ(x)Mρ
0(χ)+2fϕϕ(x).
(5.8)
Taking the limit of both sides as w→∞we have (5.6).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 12 of 22 M. Turgay, T. Acar MJOM
Theorem 8. Let χ∈ψbe a ρ-kernel with Mρ
2(χ)<∞and f◦ρ−1
ϕ◦ρ−1∈Uϕ(R),
then
lim
w→∞ Gχ,ρ
wf−fϕ=0
holds.
Proof. For functions f∈Uϕ(R), let us follow the same steps with the proof
of Theorem 7and replace δwith corresponding parameter of the uniform
continuity of f◦ρ−1
ϕ◦ρ−1. Also considering inequality (5.8)wehave
|(Gχ,ρ
wf)(x)−f(x)|
ϕ(x)≤fϕ
ϕ(x)w2Mρ
2(χ)+2|ρ(x)|fϕ
ϕ(x)wMρ
1(χ)
+εMρ
0(χ)+2fϕ
and passing to supremum in the last inequality over x∈R,wehavethe
desired result for w→∞.
Now, we give the rate of convergence of the family of operators (Gχ,ρ
w)
in terms of the weighted modulus of continuity given in (5.1).
Theorem 9. Let χ∈ψbe a ρ-kernel with Mρ
3(χ)<∞. Then for f◦ρ−1∈
Cϕ(R), we get
|(Gχ,ρ
wf)(x)−f(x)|≤9(1+|ρ(x)|)2ωϕf◦ρ−1;w−1(Mρ
0(χ)+Mρ
3(χ)) .
Proof. Using the definition of the operators Gχ,ρ
wand (5.4), we have
|(Gχ,ρ
wf)(x)−f(x)|
≤
k∈Z|χ(wρ (x)−k)|f◦ρ−1k
w−f(x)
≤9(1+ |ρ(x)|)2ωϕf◦ρ−1;δ
k∈Z|χ(wρ (x)−k)|1+ 1
δ3
k
w−ρ(x)
3
≤9(1+ |ρ(x)|)2ωϕf◦ρ−1;δMρ
0(χ)+ 1
δ3w3Mρ
3(χ)
for f◦ρ−1∈Cϕ(R)andδ≤1. Choosing δ=w−1,w ≥1, we get
|(Gχ,ρ
wf)(x)−f(x)|≤9(1+|ρ(x)|)2ωϕf◦ρ−1;w−1(Mρ
0(χ)+Mρ
3(χ))
which is the desired result.
Corollary 2. Let χ∈ψbe a ρ-kernel with Mρ
3(χ)<∞.Then,forf◦ρ−1∈
Uϕ(R),inviewof (5.2), we get
lim
w→∞ Gχ,ρ
wf−fϕ=0.
Remark 6.In Theorem 8, we stated the uniform convergence of Gχ,ρ
wfor
functions f◦ρ−1
ϕ◦ρ−1∈Uϕ(R). As a conclusion of Theorem 9, by using the prop-
erty of weighted modulus of continuity, we obtained uniform convergence of
Gχ,ρ
wfor functions f◦ρ−1∈Uϕ(R) in Corollary 2.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 13 of 22 107
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 2are 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 χ∈ψbe a ρ-kernel with Mρ
4(χ)<∞and the first order
ρ-algebraic moment of χis independent from x, i.e.,
mρ
1(χ, x)=mρ
1(χ)∈R\{0}
for every x∈R.Iffand ρexists and f
ρ∈Cϕ(R), then we have
w[(Gχ,ρ
wf)(x)−f(x)] −f(x)
ρ(x)mρ
1(χ)
≤9(1+ |ρ(x)|)2ωϕf
ρ;w−1[Mρ
1(χ)+Mρ
4(χ)]
(5.9)
at any x∈R.
Proof. By the Taylor expansion of f◦ρ−1, we can write
f◦ρ−1(ρ(u)) = f◦ρ−1(ρ(x)) + f◦ρ−1
(ρ(x)) (ρ(u)−ρ(x)) + R1(f;u, x),
where
R1(f;u, x):=f◦ρ−1(ρ(ξ)) −f◦ρ−1(ρ(x))(ρ(u)−ρ(x))
=f(ξ)
ρ(ξ)−f(x)
ρ(x)(ρ(u)−ρ(x))
(5.10)
and ξis a number between uand x. Using the above Taylor formula in the
definition of the operators Gχ,ρ
w, we obtain
(Gχ,ρ
wf)(x)=
k∈Z
χ(wρ (x)−k)f◦ρ−1k
w
=
k∈Z
χ(wρ (x)−k)f(x)+f◦ρ−1(ρ(x)) k
w−ρ(x)
+R1f;ρ−1k
w;x
:= I1+I2.(5.11)
It is clear that
I1=f(x)+f(x)
ρ(x)
1
wmρ
1(χ).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 14 of 22 M. Turgay, T. Acar MJOM
To estimate I2, if we use the inequality (5.4) and (5.10), we get
|I2|≤9(1+ |ρ(x)|)2ωϕf
ρ;δ
×
k∈Z|χ(wρ (x)−k)|
k
w−ρ(x)
+1
δ3
k
w−ρ(x)
4
≤9(1+ |ρ(x)|)2ωϕf
ρ;δ
×1
w
k∈Z|χ(wρ (x)−k)||k−wρ (x)|
+1
δ3w4
k∈Z|χ(wρ (x)−k)||k−wρ (x)|4.
Now, choosing δ=w−1,wehave
|I2|≤9(1+ |ρ(x)|)2ωϕf
ρ;w−11
wMρ
1(χ)+ 1
wMρ
4(χ).
Finally, substituting I1and I2in (5.11)wehave
w[(Gχ,ρ
wf)(x)−f(x)] −f(x)
ρ(x)mρ
1(χ)
≤9(1+ |ρ(x)|)2ωϕf
ρ;w−1[Mρ
1(χ)+Mρ
4(χ)]
which is desired result.
Corollary 3. 1. Let f∈Cϕ(R). If we choose ρ(x)=xin Theorem 10,we
have the Voronovskaja theorem obtained in [3]:
lim
w→∞ w[(Gχ,ρ
wf)(x)−f(x)] = f(x)mρ
1(χ);
2. Let f
ρ∈Uϕ(R). If we take limit of (5.9)as w→∞, we have qualitative
Voronovskaja-type theorem for Gχ,ρ
w, that is,
lim
w→∞ w[(Gχ,ρ
wf)(x)−f(x)] = f(x)
ρ(x)mρ
1(χ).
6. Examples of Some ρ-Kernels
In this section, we present examples of some ρ-kernels satisfying the assump-
tions (χ1),(χ2) and (χ3). It is well-known that using Poisson-Summation
formula given in [17], the assumption (χ2) is equivalent to
ˆχ(2πk)=1,k=0
0,k∈Z\{0},(6.1)
where ˆχ(v):=Rχ(y)e−ivy dy, v ∈R, is the Fourier transform of χ(see [20,
Lemma 4.2]).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 15 of 22 107
Central B-Spline Kernel
For n∈N, central B-splines of order nare defined by
Bn(x):= 1
(n−1)!
n
j=0
(−1)jn
jn
2+x−jn−1
+,x∈R,
where (x)n−1
+:= max xn−1,0!. The Fourier transform of Bnis
ˆ
Bn(v)=sin v
2
v
2n
,v∈R.
By considering the equality (6.1), we get
mρ
0(Bn,u)=1
for every u∈R.Since central B-splines kernels have compact supports on
−n
2,n
2, all the absolute moments of arbitrary order βof Bnare finite.
As an example, we consider the 3-order B-spline:
B3(x)=⎧
⎨
⎩
3
4−x2,|x|≤1
2,
1
23
2−|x|2,1
2<|x|≤3
2,
0,|x|>3
2
for more details, see [20].
Corollary 4. For the modified generalized sampling series with central B-
spline kernel we have
GBn,ρ
wf(x)=
k∈Zf◦ρ−1k
wBn(wρ (x)−k)
and there holds:
i. for f∈CB (R)(also for f∈Cϕ(R))
lim
w→∞ GBn,ρ
wf(x)=f(x);
ii.
lim
w→∞
GBn,ρ
wf−f
∞=0
and
lim
w→∞
GBn,ρ
wf−f
ϕ=0
for f◦ρ−1∈UC (R)and f◦ρ−1
ϕ◦ρ−1∈Uϕ(R), respectively;
iii. if f∈CB (R)
GBn,ρ
wf(x)−f(x)≤ωf◦ρ−1,w
−1(Mρ
0(Bn)+Mρ
1(Bn))
and if f◦ρ−1∈Cϕ(R)
GBn,ρ
wf(x)−f(x)
≤9(1+ |ρ(x)|)2ωϕf◦ρ−1;w−1(Mρ
0(Bn)+Mρ
3(Bn)) ;
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 16 of 22 M. Turgay, T. Acar MJOM
Figure 1. Graph of θηkernel
iv. if f∈CB (R)with fand ρexists at x∈Ror f
ρ∈Uϕ(R)with ρ
exists at x∈Rwe have
lim
w→∞ wGBn,ρ
wf(x)−f(x)=f(x)
ρ(x)mρ
1(Bn).
θη-Kernel
Let us consider the function η:R→Rgiven by
η(v):=⎧
⎪
⎨
⎪
⎩
1,v=0,
e−1
e1/v2
−e,|v|<1,x=0,
0,|v|≥1.
Then θ-kernel is defined by
θη(x)= 1
2π"1
−1
η(v)cos(xv)dv.
Again considering the equality (6.1), one can show that
mρ
0(θη,u)=1
for every u∈R. Since the ηfunction has compact support on [−1,1], θη
is a band-limited kernel. In addition, we have that θη(x)=O|x|−jas
x→±∞for all j∈N∪{0}. Then, from [11, Remark 3.2 (d)], we obtain the
absolute moments of arbitrary order βof θηare 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].
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 17 of 22 107
Figure 2. Graph of function fand operators
(Gχ
wf),(Gχ,ρ1
wf) with w= 10 and B-spline kernel
Figure 3. Graph of function gand operators (Gχ
wg),(Gχ,ρ
wg)
with w= 10 and B-spline kernel
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 ρ1:R→Rand ρ2:R→Rfunctions given by
ρ1(x):=x3+x
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 18 of 22 M. Turgay, T. Acar MJOM
Table 1. Comparison of error of approximations of the classical generalized sampling series and the modified
generalized sampling series by ρ1and ρ2for function fat some random values
w|(GB3
wf)(−1.53) −f(−1.53)||(GB3,ρ1
wf)(−1.53) −f(−1.53)||(GB3,ρ2
wf)(−1.53) −f(−1.53)|
3 0.0036159 0.0000213457 0.0000011116.
5 0.0012588. 0.0000080298. 0.0000004074.
10 0.0002854. 0.0000019613. 0.0000001040.
30 0.0000326. 0.0000002190. 0.0000000120.
50 0.0000116. 0.0000000754. 0.0000000043.
100 0.0000029. 0.0000000172. 0.0000000010.
300 0.0000003. 0.0000000002. 0.0000000001.
w|(GB3
wf)(−0.95) −f(−0.95)||(GB3,ρ1
wf)(−0.95) −f(−0.95)||(GB3,ρ2
wf)(−0.95) −f(−0.95)|
3 0.017414 0.0018106. 0.0032155
5 0.006607. 0.0006623. 0.0011454.
10 0.001661. 0.0001648. 0.0002811.
30 0.000184. 0.0000182. 0.0000317.
50 0.000066. 0.0000065. 0.0000113.
100 0.000016. 0.0000016. 0.0000028.
300 0.000001. 0.0000001. 0.0000003.
w|(GB3
wf)(1.05) −f(1.05)||(GB3,ρ1
wf)(1.05) −f(1.05)||(GB3,ρ2
wf)(1.05) −f(1.05)|
3 0.014799 0.0010999. 0.0013716.
5 0.005540. 0.0003852. 0.0004801.
10 0.001433. 0.0000958. 0.0001183.
30 0.000159. 0.0000106. 0.0000130.
50 0.000057. 0.0000038. 0.0000047.
100 0.000014. 0.0000009. 0.0000011.
300 0.000001. 0.0000001. 0.0000001.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 19 of 22 107
and
ρ2(x):=x3+2x
3.
It is easy to see that ρ1and ρ2satisfy the conditions ρ1),ρ
2)andρ3).
Now let us consider the function f:R→R,f(x)= x2
1+|x|3. Then we
have the Fig. 2.
Secondly, we consider the function g:R→R,g(x)= 1
1+(x+2)2. Then
we have the Fig. 3.
Finally, using ρ1,ρ
2and fas target function we obtain some numerical
results using central B-spline kernel of order 3 given in Table 1.
Acknowledgements
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.
Author contributions MT wrote the main manuscript text and prepared the
figures. TA analyzed the theorems and proofs in the paper.
Funding Open access funding provided by the Scientific and Technological
Research Council of T¨urkiye (T ¨
UB˙
ITAK).
Data Availability All data generated or analyzed during this study are in-
cluded in this published article. All authors reviewed the manuscript.
Declarations
Conflict of interest The authors declare no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0
International License, which permits use, sharing, adaptation, distribution and re-
production in any medium or format, as long as you give appropriate credit to
the original author(s) and the source, provide a link to the Creative Commons
licence, and indicate if changes were made. The images or other third party ma-
terial in this article are included in the article’s Creative Commons licence, unless
indicated otherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is not permitted by
statutory regulation or exceeds the permitted use, you will need to obtain permis-
sion directly from the copyright holder. To view a copy of this licence, visit http://
creativecommons.org/licenses/by/4.0/.
Publisher’s Note Springer Nature remains neutral with regard to jurisdic-
tional claims in published maps and institutional affiliations.
References
[1] Acar, T.: Asymptotic formulas for generalized Sz´asz–Mirakyan operators.
Appl. Math. Comput. 263, 233–239 (2015)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 20 of 22 M. Turgay, T. Acar MJOM
[2] Acar, T., Aral, A., Ra¸sa, I.: Positive linear operators preserving τand τ2.
Constr. Math. Anal. 2(3), 98–102 (2019)
[3] Acar, T., Alag¨oz, O., Aral, A., Costarelli, D., Turgay, M., Vinti, G.: Conver-
gence of generalized sampling series in weighted spaces. Demonstr. Math. 55,
153–162 (2022)
[4] Acar, T., Eke, A., Kursun, S.: Bivariate generalized Kantorovich-type expo-
nential sampling series, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.,
118, 35 (2024)
[5] Acar, T., Kursun, S., Turgay, M.: Multidimensional Kantorovich modifications
of exponential sampling series. Quaest. Math. 46(1), 57–72 (2023)
[6] Acar, T., Kursun, S.: Pointwise convergence of generalized Kantorovich expo-
nential sampling series. Dolomites Res. Notes Approx. 16, 1–10 (2023)
[7] Acar, T., Ulusoy, G.: Approximation by modified Sz´asz–Durrmeyer operators.
Period. Math. Hung. 72, 64–75 (2016)
[8] Alagoz, O., Turgay, M., Acar, T., Parlak, M.: Approximation by sampling dur-
rmeyer operators in weighted space of functions. Numer. Funct. Anal. Optim.
43(10), 1223–1239 (2022)
[9] Angeloni, L., Cetin, N., Costarelli, D., Sambucini, A.R., Vinti, G.: Multivari-
ate sampling Kantorovich operators: quantitative estimates in Orlicz spaces.
Constr. Math. Anal. 4(2), 229–241 (2021)
[10] Aral, A., Acar, T., Kursun, S.: Generalized Kantorovich forms of exponential
sampling series. Anal. Math. Phys. 12, 50 (2022)
[11] 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)
[12] Bardaro, C., Faina, L., Mantellini, I.: A generalization of the exponential sam-
pling series and its approximation properties. Math. Slovaca 67(6), 1481–1496
(2017)
[13] Bardaro, C., Mantellini, I.: Asymptotic formulae for linear combinations of
generalized sampling type operators. Zeitschrift f¨ur Anal. und ihre Anwendung
32, 279–298 (2013)
[14] Bardaro, C., Mantellini, I.: Asymptotic expansion of generalized Durrmeyer
sampling type series. Jaen J. Approx. 6(2), 143–165 (2014)
[15] Bernstein, S.N.: D´emonstration du th´eor`eme de Weierstrass, fond´ee sur le calcul
des probabilit´es., Math. Charkow (2), 13(1-2), in French (1912)
[16] 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)
[17] Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation I. Academic
Press, New York-London (1971)
[18] Butzer, P.L., Schmeisser, G., Stens, R.L.: Basic relations valid for the Bernstein
space Bpand their extensions to functions from larger spaces with error esti-
mates in term of their distances from Bp. J. Fourier Anal. Appl. 19, 333–375
(2013)
[19] Butzer, P.L., Splettstosser, W.: A sampling theorem for duration-limited func-
tions with error estimates. Inf. Control. 34(1), 55–65 (1977)
[20] 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)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
MJOM Approximation by Modified... Page 21 of 22 107
[21] 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)
[22] Butzer, P.L., Stens, R.L.: The sampling theorem and linear prediction in signal
analysis. Jahresber. Dtsch. Math. Ver. 90(1), 1–70 (1998)
[23] C´ardenas-Morales, D., Garrancho, P., Ra¸sa, I.: Bernstein-type operators which
preserve polynomials. Comput. Math. with Appl. 62, 158–163 (2011)
[24] 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)
[25] Costarelli, D., Vinti, G.: Rate of approximation for multivariate sampling Kan-
torovich operators on some functions spaces. J. Integr. Equ. Appl. 26(4), 455–
481 (2014)
[26] 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)
[27] DeVore, R.A., Lorentz, G.G.: Constructive approximation, vol. 303. Springer,
Berlin (1993)
[28] Draganov, B.R.: A fast converging sampling operator. Constr. Math. Anal.
5(4), 190–201 (2022)
[29] Erencin, A., Ra¸sa, I.: Voronovskaya type theorems in weighted spaces. Numer.
Funct. Anal. Optim. 37(12), 1517–1528 (2016)
[30] Holhos, A.: Quantitative estimates for positive linear operators in weighted
space. Gen. Math. 16(4), 99–110 (2008)
[31] Ilarslan, H.G.I., Ba¸scanbaz-Tunca, G.: Convergence in variation for Bernstein-
type operators. Mediterr. J. Math. 13(5), 2577–2592 (2016)
[32] King, J.P.: Positive linear operators which preserve x2. Acta. Math. Hungar.
99, 203–208 (2003)
[33] Kotel’nikov, V.A.: On the carrying capacity of “ether” and wire in electrocom-
munications, Material for the First All-Union Conference on the Questions of
Communications, Moscow (1933)
[34] Acar T., Kursun S., Acar O.: Approximation properties of exponential sam-
pling series in logarithmic weighted spaces. B. Iran Math. Soc. 50, 36 (2024).
[35] Kursun, S., Aral, A., Acar, T.: Approximation results for Hadamard-type ex-
ponential sampling Kantorovich series. Mediterr. J. Math. 20, 263 (2023)
[36] Kursun, S., Aral, A., Acar, T.: Riemann-Liouville fractional integral type expo-
nential sampling Kantorovich series, Exp. Syst. Appl., 238(F), 122350 (2024)
[37] Kursun, S., Turgay, M., Alagoz, O., Acar, T.: Approximation Properties of
Multivariate Exponential Sampling Series. Carpathian Math. Publ. 13(3), 666–
675 (2021)
[38] ¨
Ozer, 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)
[39] Riesz, S., Stens, R.L.: Approximation by generalized sampling series, Bl.
Sendov et al., Publ. House Bulgarian Acad. Sci (Sofia), 746–756 (1984)
[40] Shannon, C.E.: Communications in the presence of noise. Proc. IRE. 37, 10–21
(1949)
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
107 Page 22 of 22 M. Turgay, T. Acar MJOM
[41] Whittaker, E.T.: On the functions, which are represented by expansions of the
interpolation theory. Proc. R. Soc. Edinb. 35, 181–194 (1915)
Metin Turgay and Tuncer Acar
Department of Mathematics, Faculty of Science
Selcuk University
Selcuklu, Konya 42003
Turk e y
e-mail: metinturgay@yahoo.com
Tuncer Acar
e-mail: tunceracar@ymail.com
Received: March 12, 2024.
Revised: April 8, 2024.
Accepted: April 10, 2024.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
1.
2.
3.
4.
5.
6.
Terms and Conditions
Springer Nature journal content, brought to you courtesy of Springer Nature Customer Service Center
GmbH (“Springer Nature”).
Springer Nature supports a reasonable amount of sharing of research papers by authors, subscribers
and authorised users (“Users”), for small-scale personal, non-commercial use provided that all
copyright, trade and service marks and other proprietary notices are maintained. By accessing,
sharing, receiving or otherwise using the Springer Nature journal content you agree to these terms of
use (“Terms”). For these purposes, Springer Nature considers academic use (by researchers and
students) to be non-commercial.
These Terms are supplementary and will apply in addition to any applicable website terms and
conditions, a relevant site licence or a personal subscription. These Terms will prevail over any
conflict or ambiguity with regards to the relevant terms, a site licence or a personal subscription (to
the extent of the conflict or ambiguity only). For Creative Commons-licensed articles, the terms of
the Creative Commons license used will apply.
We collect and use personal data to provide access to the Springer Nature journal content. We may
also use these personal data internally within ResearchGate and Springer Nature and as agreed share
it, in an anonymised way, for purposes of tracking, analysis and reporting. We will not otherwise
disclose your personal data outside the ResearchGate or the Springer Nature group of companies
unless we have your permission as detailed in the Privacy Policy.
While Users may use the Springer Nature journal content for small scale, personal non-commercial
use, it is important to note that Users may not:
use such content for the purpose of providing other users with access on a regular or large scale
basis or as a means to circumvent access control;
use such content where to do so would be considered a criminal or statutory offence in any
jurisdiction, or gives rise to civil liability, or is otherwise unlawful;
falsely or misleadingly imply or suggest endorsement, approval , sponsorship, or association
unless explicitly agreed to by Springer Nature in writing;
use bots or other automated methods to access the content or redirect messages
override any security feature or exclusionary protocol; or
share the content in order to create substitute for Springer Nature products or services or a
systematic database of Springer Nature journal content.
In line with the restriction against commercial use, Springer Nature does not permit the creation of a
product or service that creates revenue, royalties, rent or income from our content or its inclusion as
part of a paid for service or for other commercial gain. Springer Nature journal content cannot be
used for inter-library loans and librarians may not upload Springer Nature journal content on a large
scale into their, or any other, institutional repository.
These terms of use are reviewed regularly and may be amended at any time. Springer Nature is not
obligated to publish any information or content on this website and may remove it or features or
functionality at our sole discretion, at any time with or without notice. Springer Nature may revoke
this licence to you at any time and remove access to any copies of the Springer Nature journal content
which have been saved.
To the fullest extent permitted by law, Springer Nature makes no warranties, representations or
guarantees to Users, either express or implied with respect to the Springer nature journal content and
all parties disclaim and waive any implied warranties or warranties imposed by law, including
merchantability or fitness for any particular purpose.
Please note that these rights do not automatically extend to content, data or other material published
by Springer Nature that may be licensed from third parties.
If you would like to use or distribute our Springer Nature journal content to a wider audience or on a
regular basis or in any other manner not expressly permitted by these Terms, please contact Springer
Nature at
onlineservice@springernature.com
