KYUNGPOOK Math. J. 2019; 59(3): 465-480  
Approximation by Generalized Kantorovich Sampling Type Series
Angamuthu Sathish Kumar∗, Ponnaian Devaraj
Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, Nagpur-440010, India
e-mail : mathsatish9@gmail.com
Department of Mathematics, Indian Institute of Science Education and Research, Thiruvananthapuram, India
e-mail : jujeong@deu.ac.kr
Received: November 10, 2017; Accepted: January 28, 2019; Published online: September 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

In the present article, we analyse the behaviour of a new family of Kantorovich type sampling operators (Kwφf)w>0. First, we give a Voronovskaya type theorem for these Kantorovich generalized sampling series and a corresponding quantitative version in terms of the first order of modulus of continuity. Further, we study the order of approximation in C(ℝ), the set of all uniformly continuous and bounded functions on ℝ for the family (Kwφf)w>0. Finally, we give some examples of kernels such as B-spline kernels and the Blackman-Harris kernel to which the theory can be applied.

Keywords: sampling Kantorovich operators, Voronovskaya type formula, rate of convergence, modulus of smoothness.
Introduction
roduction

The theory of generalized sampling series was first initiated by P. L. Butzer and his school [13] and [14]. In recent years, it has become an attractive topic in approximation theory due to its wide range of applications, especially in signal and image processing. For w > 0, a generalized sampling series of a function f : ℝ → ℝ is defined by (Twφf)(x)=k=φ(wxk)f(kw),x,where φ is a kernel function on ℝ. These type of operators have been studied by many authors (e.g. [7, 9, 11, 23, 24, 25, 26, 27]).

The study of Kantorovich type generalizations of approximation operators is an important subject in approximation theory, as they can be used to approximate Lebesgue integrable functions. In the last few decades, Kantorovich modifications of several operators have been constructed and their approximation behavior studied, we mention some of the work in this direction e.g., [1, 2, 3, 22] etc.

In [12], the authors introduced the sampling Kantorovich operators and studied their rate of convergence in the general settings of Orlicz spaces. After that, Danilo and Vinti [16] extended their study in the multivariate setting and obtained the rate of convergence for functions in Orlicz spaces. Danilo and Vinti [18], obtained the rate of approximation for the family of sampling Kantorovich operators in the uniform norm, for bounded uniformly continuous functions belonging to Lipschitz classes and for functions in Orlicz spaces. Also, the nonlinear version of sampling Kantorovich operators have been studied in [17] and [29].

Altomare and Leonessa [5] considered a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. Such operators include the Kantorovich operators as a particular case. Later, in order to obtain an approximation process for spaces of locally integrable functions on unbounded intervals, Altomare et. al. introduced and studied the generalized Szász-Mirakjan-Kantorovich operators in [4]. Also in [15], the authors obtained some qualitative properties and an asymptotic formula for such a sequence of operators.

We consider the generalized Kantorovich type sampling series. Let {ak}k∈ℤ and {bk}k∈ℤ be two sequences of real numbers such that bkak = ∆k > 0 for k ∈ ℤ. In this paper, we analyse the approximation properties of the following type of generalized Kantorovich sampling series (Kwφf)(x)=k=wΔkφ(wxk)k+akwk+bkwf(u)du,where f is in C(ℝ), the class of all uniformly continuous and bounded functions on ℝ.

In the present paper, first we obtain asymptotic formula and their quantitative estimate for the operators (Kwφf)w>0. Further, the order of approximation of these operators is analysed in C(ℝ) Finally, we give some examples of kernels such as B-spline kernels and the Blackman-Harris kernel, to which the theory can be applied.

Main Results
n Results

Let φC(ℝ) be fixed. For every ν ∈ ℕ0 = ℕ ∪{0}, u ∈ ℝ we define the algebraic moments as mν(φ,u):=k=ϕ(uk)(ku)νand the absolute moments by Mν(φ):=supuk=|φ(uk)||(ku)|ν.

Note that for μ, νN0 with μ < ν, Mν(φ) < + implies Mμ(φ) < +. Indeed for μ < ν, we have k=|φ(uk)||(ku)|ν=|uk|<1|φ(uk)||(ku)|ν+|uk|1|φ(uk)||(ku)|ν2φ+|uk|1|φ(uk)||(ku)|ν|(ku)|νμ2φ+Mν(φ).When φ is compactly supported, we immediately have that Mν(φ) < + for every ν ∈ ℕ0.

We suppose that the following assumptions hold:

  • for every u ∈ ℝ we have k=φ(uk)=1,

  • M2(φ) < + and there holds limr|uk|>r|φ(uk)|(ku)2=0uniformly with respect to u ∈ ℝ

  • for every u ∈ ℝ, we have m1(φ,u)=m1(φ)=k=φ(uk)(ku)=0,

  • supk{|ak|,|bk|}=M*<.

Theorem 2.1

Let fC(ℝ) and {ak} and {bk} be two bounded sequences of real numbers such that ak +bk = α and bkak > ∆ > 0. If f′(x) exits at a point x ∈ ℝ then, limww[(Kwφf)(x)f(x)]=α2f(x).

Proof

Let M* = supk{|ak|, |bk|}. From the Taylor’s theorem, we have f(u)=f(x)+f(x)(ux)+h(ux)(ux),for some bounded function h such that h(t) 0 as t → 0.

Thus, we have (Kwφf)(x)f(x)=f(x)k=wΔkφ(wxk)k+akwk+bkw(ux)du+k=wΔkφ(wxk)k+akwk+bkwh(ux)(ux)du=I1+I2,(say).First, we obtain I1. I1=f(x)k=wbkakφ(wxk)k+akwk+bkw(ux)du=f(x)2k=wbkakφ(wxk)[(k+bkwx)2(k+akwx)2]=f(x)2wk=φ(wxk)[(bk+ak)+2(kwx)]=αf(x)2wk=φ(wxk)+f(x)wk=φ(wxk)(kwx)=af(x)2w.Next, we obtain I2. I2=k=wbkakφ(wxk)k+akwk+bkwh(ux)(ux)du.In order to obtain an estimate of I2, let ∊ > 0 be fixed. Then, there exists δ > 0 such that |h(t)| ≤ ∊ for |t| ≤ δ. Then, we have |I2||kwx|<δw/2wbkak|φ(wxk)|k+akwk+bkw|h(ux)||ux|du+|kwx|δw/2wbkak|φ(wxk)|k+akwk+bkw|h(ux)||(ux)|du=J1+J2,(say).First, we estimate J1. We have J1|kwx|<δw/2wbkak|φ(wxk)|k+akwk+bkw|(ux)|du2k=wbkak|φ(wxk)|[(k+bkwx)2+(k+akwx)2]wΔ*((M*)2M0(φ)+2M*M1(φ)+M2(φ)).Next, we obtain J2. Let R > 0 be such that |uk|>R|φ(uk)|(uk)2<uniformly with respect to u ∈ ℝ. Also, let w be such that δw/2 > R. Then |kwx|>δw/2|φ(wxk)|(wxk)2<for every x ∈ ℝ. The same inequality holds also for the series |kwx|>δw/2|φ(wxk)|(wxk)j<for j = 0, 1. Hence, we get J2h|kwx|δw/2wbkak|φ(wxk)|k+akwk+bkw|(ux)|duh2|kwx|δw/2wbkak|φ(wxk)|[(k+bkwx)2+(k+akwx)2]hwΔ*(1+M*)2.Hence, the proof is completed.

Remark 2.1

The boundedness assumption on f can be relaxed by assuming that there are two positive constant a, b such that |f(x)| ≤ a + b|x|, for every x ∈ ℝ. We have |Kwφf)(x)|k=wbkak|φ(wxk)|k+akwk+bkw|f(u)|duk=wbkak|φ(wxk)|k+akwk+bkw(a+b|u|)duM0(φ)(a+b|x|)|+bΔ*w(M2(φ)+2(M*)M1(φ)+(M*)2M0(φ))and hence the series Kwφf is absolutely convergent for every x ∈ ℝ Moreover, for a fixed x0 ∈ ℝ, P1(x)=f(x0)+f(x0)(xx0,)the Taylor’s polynomial of first order centered at the point x0, by the Taylor’s formula we can write f(x)P1(x)(xx0)=h(xx0),where h is a function such that limt0h(t)=0. Then h is bounded on [x0δ, x0 + δ], for some δ > 0. For |xx0| > δ, we have |h(xx0)|a+b|x||xx0|+|P1(x)||xx0|a+b|x||xx0|+|f(x0)||xx0|+|f(x0)|,and the terms on the right-hand side of the above inequality are all bounded for |xx0| > δ. Hence, h(. − x0) is bounded on ℝ. Along the lines of the proof of Theorem 2.1., the same Voronovskaya formula can be obtained.

Let Cm denote the set of all fC(ℝ) such that f is m times continuously differentiable and ‖f(m)< .

Let δ > 0. For fC(ℝ), the Peetre’s K-functional is defined as K(δ,f,C,C1):=inf{fg+δg:gC1}.For a given δ > 0, the usual modulus of continuity of a given uniformly continuous function f : ℝ → ℝ is defined as ω(f,δ):=sup|xy|δ|f(x)f(y)|.It is well known that, for any positive constant λ > 0, the modulus of continuity satisfies the following property ω(f,λδ)(λ+1)ω(f,δ).For a function fCm, x0, x ∈ ℝ and m ≥ 1, the Taylor’s formula is given by f(x)=k=0mf(x)(x0)k!(xx0)k+Rm(f;x0,x)and the remainder term Rm(f; x0, x) is estimated by |Rm(f;x0,x)||xx0|mm!ω(f(m);|xx0|).For every fC(ℝ) there holds K(δ/2,f,C,C1)=12ω¯(f,δ),where ω̄(f, .) denotes the least concave majorant of ω(f, .) (see e.g. [6]).

The following estimate for the remainder Rm(f; x0, x) in terms of ω̄ was proved in [21].

Lemma 2.1

Let fCm, m ∈ ℕ0and x0, x ∈ ℝ Then, we have|Rm(f;x0,x)||xx0|mm!ω¯(f(m);|xx0|m+1).

We have the following quantitative version of Theorem 2.1 in terms of the modulus ω̄ in case of m = 1.

Theorem 2.2

Let fC1and {ak} and {bk} be two sequences of real numbers such that ak+ bk = α, bkak≥ ∆* > 0 and supk {|ak|, |bk|} ≤ M*. Then, for very x ∈ ℝ, the following hold:|w[(Kwφf)(x)f(x)]αf(x)2|AΔ*ω¯(f,Δ*2w)where A = (M*)2M0(φ) + 2(M*)M1(φ) + M2(φ).

Proof

Let fC1 be fixed. Then, we can write |w[(Kwφf)(x)f(x)]αf(x)2|=|f(x)k=w2bkakφ(wxk)k+akwk+bkw(ux)du+k=w2bkakφ(wxk)k+akwk+bkwh(ux)(ux)duαf(x)2|k=w2bkak|φ(wxk)|k+akwk+bkw|h(ux)||(ux)|du.Using the relation (2.2) and Lemma 2.1, we obtain |w[(Kwφf)(x)f(x)]αf(x)2|k=w2bkak|φ(wxk)|k+akwk+bkw|(ux)|ϖ(f,|xu|2)du=2k=w2bkak|φ(wxk)|k+akwk+bkw|(ux)|K(|ux|4,f,C,C1)du:=I1.For gC2, we have I1k=2w2bkqk|φ(wxk)|k+akwk+bkw|(ux)|((fg)+|ux|4g)du(fg)k=2w2bkak|φ(wxk)|k+bkwk+bkw|(ux)|du+gk=2w2bkak|φ(wxk)|k+akwk+bkw(ux)2du(fg)k=|φ(wxk)|w2bkak[(k+bkwx)2+(k+akwx)2]+gk=w26(bkak)|φ(wxk)|[(k+bkwx)3(k+akwx)3](fg)Δ*(2(M*)2M0(φ)+4M1(φ)(M*)+2M2(φ))+g16w(3(M*)2M0(φ)+6(M*)M1(φ)+3M2(φ))(fg)2Δ*((M*)2M0(φ)+2M1(φ)(M*)+M2(φ))+g12w((M*)2M0(φ)+2M1(φ)(M*)+M2(φ))2AΔ*((fg)+gΔ*4w).Taking the infimum over all gC2, we get I1AΔ*ω¯(f,Δ*2w).Hence, the proof is completed.

Remark 2.2

As a consequence of Theorem 2.2, under the above assumptions we get the uniform convergence for w[(Kwφf)(x)f(x)] to α2f(x).

Remark 2.3

Note that when φ is supported in I = [−R, R], R > 0 we can obtain a different estimate for I1. |w[(Kwφf)(x)f(x)]α2f(x)|M0(φ)(R2+2RM*+(M*)2)Δ*ω¯(f,Δ2w).Also, we obtain I12M0(φ)(R2+2RM*+(M*)2)Δ*[(fg)+gΔ*4w].

Next, we study the order of approximation for the generalized sampling Kantorovich series. The order of approximation for the generalized sampling Kantorovich series has been extensively studied by many authors (see [10, 18, 19, 20]).

Theorem 2.3

Let φ be a kernel satisfying an additional condition thatMβ(φ)=supuk|φ(uk)||(ku)|β<+for some 0 < β < 1 and {ak} and {bk} be two bounded sequences of real numbers. Then, for any fC(ℝ) , we have|(Kwφf)(x)f(x)|ω(f,wβ)(Mβ(φ)+(M*)βM0(φ)+M0(φ))+2β+1fwβMβ(φ),for every x ∈ ℝ and w >2M*.

Proof

Let x ∈ ℝ be fixed. Then, for w > 0, we can write |(Kwφf)(x)f(x)|=|(Kwφf)(x)f(x)kZφ(wxk)|kZ(wbkakk+akwk+bkw|f(u)f(x)|du)|φ(wxk)|=:J.Now, we estimate J. J|wxk|w/2(wbkakk+akwk+bkw|f(u)f(x)|du)|φ(wxk)|+|wxk|>w/2(wbkakk+akwk+bkw|f(u)f(x)|du)|φ(wxk)|=:I1+I2.We observe that, for every u[k+akw,k+bkw] and suitable large w with |wxk| ≤ w/2, we get |uw||u(k+akw)+(k+akw)x||ukw|+|kwx|M*w+121,for every w ≥ 2M*. Since 0 < β < 1, we have ω(f,|ux|)ω(f,|ux|β).Using the property of modulus of continuity (2.1), we obtain I1|wxk|w/2(wbkakk+akwk+bkwω(f,|ux|β)du)|φ(wxk)||wxk|w/2(wbkakk+akwk+bkw(1+wβ|ux|β)ω(f,wβ)du)|φ(wxk)|ω(f,wβ)[|wxk|w/2(wbkakk+akwk+bkwwβ|ux|βdu)|φ(wxk)+|wxk|/2|φ(wxk)||]=:ω(f,ωβ)(J1+J2).First, we obtain J1. Using the property of sub-addivity of |.|β with 0 < β < 1, we have J1|wxk|w/2(wβmaxu[k+akw,k+bkw]|ux|β)|φ(wxk)||wxk|w/2(wβmax(|k+akwx|β,|k+bkwx|β))|φ(wxk)||wxk|w/2(wβmax(|kwx|β+|akw|β,|kwx|β+|bkw|β))|φ(wxk)||wxk|/2wβ(|kwx|β+(supk{|ak|,|bk|})βwβ)|φ(wxk)||wxk|w/2|kwx|β|φ(wxk)|+|wxk|w/2(M*)β|φ(wkk)|Mβ(φ)+(M*)βM0(φ)<.It is easy to see that J2|wxk|w/2|φ(wxk)|=M0(φ).Next, we estimate I2. I22f|wxk|>w/2(wbkakk+akwk+bkwdu)|φ(wxk)|2f|wxk|>w/2|φ(wxk)|2f|wxk|>w/2|wxk|β|wxk|β|φ(wxk)|2fwβ|wxk|>w/2|wxk|β|φ(wxk)|2β+1fwβMβ(φ)<+,which completes the proof.

Applications to Special Kernels
l Kernels

In this section, we describe some particular examples of kernels φ which illustrates the previous theory. In particular, we will examine the B-splines kernel and Blackman-Harries kernel.

Combinations of B-spline functions

First, we consider the sampling Kantorovich operators based upon the combinations of spline functions. For h ∈ ℕ, the B–spline of order h is defined as Bh(x):=χ[12,12]*χ[12,12]*χ[12,12]**χ[12,12],(htimes)where χ[12,12]={1,if12x120,otherwiseand * denotes the convolution. The Fourier transform of the functions Bh(x) is given by B^h(w)=(χ[12,12]^(w))h=(sinw/2w/2)h,w,h(see [14] and [28]). Given real numbers ∊0, ∊1 with ∊0 < ∊1 we will construct the linear combination of translates of Bh, with h ≥ 2 of type φ(x)=a0Bh(x0)+a1Bh(x1).The Fourier transform of φ is given by φ^(w)=(a0ei0w+a1ei1w)B^h(w).Using the Poisson summation formula (i)jk=φ(uk)(uk)j~k=φ^(j)(2πk)ei2πku,we obtain k=φ(uk)=k=φ^(2πk)ei2πku.We have Bh^(2πk)=(sin(πk)πk)h={1,ifk=00,ifk0and hence φ^(2πk)={a0,a1ifk=00,ifk0.Thus k=φ(uk)=a0+a1.Therefore, condition (i) is satisfied if a0 + a1 = 1. Now, we show that condition (iii) is also satisfied. Again from the Poisson summation formula, we obtain (i)k=φ(uk)(uk)=k=φ^(2πk)ei2πku.Also, we have φ^(w)=(i0a0ei0wi1a1ei1w)B^h(w)+(a0ei0w+a1ei1w)B^h'(w)Since B^h'(2πk)=0, ∀k which implies that φ̂′ (2πk) = 0. Thus, we have φ^(0)=a0+a1=1,φ^(0)=0a0+1a1=0.Solving the above linear system we get the unique solution a0=110,a1=001.Moreover it is easy to see that the support of the function φ is contained in the interval [0h2,1h2,]. Since φ(uk) = 0 if |uk| > r for r sufficiently large, we have limr|ku|>rφ(uk)(ku)2=0.Condition (ii) is satisfied. Finally, we verify the condition that Mβ(φ) <∞. k|φ(uk)||ku|β=|ku|<R|φ(uk)||ku|β+|ku||φ(uk)||ku|β.We can see that supu|{k:|uk|<R}|N0. Thus, we get Mβ(φ)=k|φ(uk)||ku|β<.

A particular Blackman-Harris kernel

Next, we consider the Blackman-Harris kernel. For every x ∈ ℝ we define the kernel (see [8]) φ(x)H(x)12sinc(x)+932(sinc(x+1)+sinc(x1))132(sinc(x+3)+sinc(x3)),where sinc(x)=sinπxπx. From [24], there holds that H(x) = O(|x|−5) as |x| → ∞. In view of [12], it follows that M2(H) is finite and limr|ku|>r|H(uk)|(uk)2=0.Indeed, there exists N0 > 0 such that |H(x)| ≤ M/|x|5 for |x| ≥ N0. Thus, we have for r > N0|ku|>r|H(uk)|(uk)2M|ku|>r1|uk|3Mr|ku|>r1|uk|22Mrk=11k2.The Fourier transform of the function H(x) is given by H^(w)=12πλ(wπ),where λ(w)=(12+916cos(πw)116cos(3πw))χ[1.1](w), χI is the characteristic function of the set I. From Lemma 3 in [14], we obtain m1(H)=k=H(uk)(uk)=0.Hence the condition (i)–(iii) are satisfied. Finally, we verify that Mβ(φ) < ∞. k|H(uk)||ku|β=|ku|<R|H(uk)||ku|β+|ku|R|H(uk)||ku|β=S1+S2,(say).First, we consider S2. There exists N > 0 such that |H(x)| ≤ M/|x|5 for |x| ≥ N. Thus, we have for R > N, S2M|ku|R|uk|β|uk|52Mk=11k5β.Next, we estimate S1. We have supu|{k:|uk|<R}|<. Thus, we obtain S1|ku|<R|H(uk)||ku|β|ku|<RM0Rβ2RM0Rβ,where ⌈x⌉ denotes the smallest integer greater than or equal to x. Hence, we get Mβ(H)=k|H(uk)||ku|β<.Thus, all the conditions are satisfied for the function H(x).

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