In the present paper, we introduce a modification of Ismail-May operators having weights of Baskakov basis functions. We estimate weighted Korovkin's theorem and difference estimates between two operators and establish a Voronovskaja type asymptotic formula in simultaneous approximation

Michael Th. Rassias

Vijay Gupta

Directory of Open Access Journals

JOURNAL OF NUMERICAL ANALYSIS AND APPROXIMATION THEORYJ. Numer. Anal. Approx. Theory, vol. 50 (2021) no. 2, pp. 153–163ictp.acad.ro/jnaatASYMPTOTIC FORMULA IN SIMULTANEOUS APPROXIMATIONFOR CERTAIN ISMAIL-MAY-BASKAKOV OPERATORSVIJAY GUPTA∗ and MICHAEL TH. RASSIAS†Abstract. The present study deals with a modification of Ismail-May operatorswith weights of Beta basis functions. We estimate weighted Korovkin’s theoremand difference estimates between two operators and establish an asymptotic for-mula for the derivative of a function.MSC 2010. 41A25, 41A30.Keywords. Ismail-May operators, Baskakov basis, asymptotic formula, simul-taneous approximation, weighted Korovkin’s theorem.1. THE OPERATORIn 1978 Ismail and May [13] (see also [12]) studied certain operators of ex-ponential type and introduced some new exponential operators. The followingsequence of operators constitutes one of the families of linear positive operatorproposed in [13, (3.14)]:(1) (Rnf)(x) =∞∑ν=0rn,ν(x)Fn,ν ,where Fn,ν = f(νn)andrn,ν(x) = e−(n+ν)x/(1+x) n(n+ν)ν−1ν!(x1+x)ν.These operators are exponential type operators asx(1 + x)2[(Rnf)(x)]′ =∞∑ν=0(ν − nx)rn,ν(x)f(νn).∗Department of Mathematics, Netaji Subhas University of Technology, Sector 3 Dwarka,New Delhi 110078, India, e-mail: vijaygupta2001@hotmail.com.†Department of Mathematics and Engineering Sciences, Hellenic Military Academy, 16673Vari Attikis, Greece,†Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Institutskiy per, d. 9,Russia,†Institute for Advanced Study, Program in Interdisciplinary Studies, 1 Einstein Dr, Prince-ton, NJ 08540, USA, e-mail: mthrassias@yahoo.com.154 Vijay Gupta and M. T. Rassias 2In recent studies, Gupta and Rassias [10] presented a collection of the momentsof several linear positive operators, Agratini [2] estimated approximation prop-erties of integral-type operators. The operators Rn are not suitable in orderto approximate integral functions. Additionally, it is observed in [6] that ob-taining the usual Durrmeyer variant of the operators (1) is not possible dueto the non-convergence of the integral∫ ∞0rn,ν(u)du.In the recent work [6], the hybrid link operators were introduced and theapproximation behavior was discussed. We consider here new hybrid operatorswith weights bn+2,ν−1(t) of slightly modified Baskakov basis functions, whichfor x ∈ [0,∞) are defined as(Vnf)(x) =∞∑ν=0rn,ν(x)Gn,ν(f),(2)whereGn,ν(f) ={(n+ 1)∫∞0 bn+2,ν−1(u)f(u)du : 1 ≤ ν <∞f(0) : ν = 0bn,ν(u) =(n+ν−1ν)uν(1+u)n+ν .For simplicity, we use the notationvn,ν(u) = (n+ 1)bn+2,ν−1(u)throughout the paper.The operators (2) reproduce linear functions. The investigation of conver-gence of operators in ordinary and simultaneous approximation has been anactive area of research over recent years. We mention here some of the relatedwork viz. [3], [7], [8], [9], [11]. In the present paper, we estimate direct resultin weighted norm, difference of these operators with discrete operators andprove a formula of asymptotic kind for derivatives of the operators Vn.2. SOME LEMMASLemma 1. IfUn,q(x) =∞∑ν=0rn,ν(x)(νn − x)q, q ∈ N ∪ {0},then the moments satisfy the relation:nUn,q+1(x) = x(1 + x)2[(Un,q(x))′ + qUn,q−1(x)].Also, one can observe thatUn,q(x) = Ox(n−[(q+1)/2]),where [s] stands for the integral part of s.3 Ismail-May-Baskakov operators 155Furthermore, for ei(t) = ti, i = 0, 1, 2, . . ., we haven(Rneq+1)(x) = x(1 + x)2[(Rneq)(x)]′ + nx(Rneq)(x).Lemma 2. If we denote µn,q(x) = (Vn(e1 − xe0)q)(x) for q ∈ N and x ≥ 0,then(n− q)µn,q+1(x) = x(1 + x)2[[µn,q(x)]′ + qµn,q−1(x)]+qµn,q(x) + qx(1 + x)µn,q−1(x).In particular, we haveµn,0(x) = 1, µn,1(x) = 0, µn,2(x) = x(1+x)(2+x)n−1 .In general for all x ∈ [0,∞), we have µn,q(x) = Ox(n−[(q+1)/2]).Proof. Usingx(1 + x)2r′n,ν(x) = (ν − nx)rn,ν(x)andu(1 + u)v′n,ν(u) = [(ν − 1)− (n+ 2)u]vn,ν(u),we havex(1 + x)2[µn,q(x)]′ ==∞∑ν=1x(1 + x)2r′n,ν(x)∫ ∞0vn,ν(u)(e1 − xe0)qdu+ne−nx/(1+x)(−x)q+1 + q(−x)q(1 + x)2e−nx/(1+x)−qx(1 + x)2∞∑ν=1rn,ν(x)∫ ∞0vn,ν(u)(e1 − xe0)q−1du=∞∑ν=1(ν − nx)rn,ν(x)∫ ∞0vn,ν(u)(e1 − xe0)qdu+ne−nx/(1+x)(−x)q+1 − qx(1 + x)2µn,q−1(x)=∞∑ν=1rn,ν(x)∫ ∞0u(1 + u)v′n,ν(u)(e1 − xe0)qdu+ (n+ 2)[µn,q+1(x)−(−x)q+1e−nx/(1+x)] + (1 + 2x)[µn,q(x)− (−x)qe−nx/(1+x)]+ne−nx/(1+x)(−x)q+1 − qx(1 + x)2µn,q−1(x).Thus, we havex(1 + x)2[[µn,q(x)]′ + qµn,q−1(x)]==∞∑ν=1rn,ν(x)∫ ∞0[(e1−xe0)2+(1+2x)(e1−xe0)+x(1+x)]v′n,ν(u)(e1−xe0)qdu+(n+ 2)[µn,q+1(x)− (−x)q+1e−nx/(1+x)]+(1 + 2x)[µn,q(x)− (−x)qe−nx/(1+x)] + ne−nx/(1+x)(−x)q+1.156 Vijay Gupta and M. T. Rassias 4Integrating by parts the last integral, we obtain thatx(1 + x)2[[µn,q(x)]′ + qµn,q−1(x)]= −(q + 2)µn,q+1(x)− (q + 1)(1 + 2x)µn,q(x)− qx(1 + x)µn,q−1(x)+(n+ 2)µn,q+1(x) + (1 + 2x)µn,q(x).This completes the proof of the recurrence relation. The other consequencesfollow from the recurrence relation. Lemma 3. If we denote Tn,q(x) = (Vneq)(x) for q ∈ N and x ≥ 0, then(n− q)Tn,q+1(x) = x(1 + x)2[Tn,q(x)]′ + (q + nx)Tn,q(x).In general, we haveTn,q(x) = xq + q(q−1)2 xq−1 (1+x)(2+x)n−q+1 +Ox(n−2).The proof of the lemma follows along the lines of Lemma 1, we thus omitthe details.Lemma 4 ([6]). There exist the polynomials φi,j,q(x) independent of n andν such that[xq(1 + x)2q] ∂q∂xq [rn,ν(x)] =∑2i+j≤qi,j≥0ni(ν − nx)jφi,j,q(x)[rn,ν(x)].Lemma 5. If µJn,νr =∑ri=0(ri)(−1)iJn,ν(er−i)[Jn,ν(e1)]i, then we haveµFn,νm = 0,m = 1, 2, 3, . . . , µGn,ν2 = ν2+nνn2(n−1) , µGn,ν3 = 4ν3+6nν2+2n2νn3(n−1)(n−2)andµGn,ν4 =3(6+n)ν4+6(6+n)nν3+3(8+n)n2ν2+6n3νn4(n−1)(n−2)(n−3) .Proof. By definition, we havebFn,ν = Fn,ν(e1) = νn .By simple computation, we getµFn,νm = 0,m = 1, 2, 3, . . .MoreoverGn,ν(eq) = (n+ 1)∫ ∞0bn+2,ν−1(u)uqdu= (n+ 1)(n+νν−1) ∫ ∞0uν+q−1(1+u)n+ν+1du =(ν+q−1)!(n−q)!(ν−1)!n!ThusbGn,ν = Gn,ν(e1) = νn ,andµGn,ν2 =2∑i=0(2i)(−1)iGn,ν(e2−i)[Gn,ν(e1)]i5 Ismail-May-Baskakov operators 157= Gn,ν(e2)− [Gn,ν(e1)]2= ν2+nνn2(n−1) .µGn,ν3 =3∑i=0(3i)(−1)iGn,ν(e3−i)[Gn,ν(e1)]i= Gn,ν(e3)− 3Gn,ν(e2)Gn,ν(e1) + 3[Gn,ν(e1)]3 − [Gn,ν(e1)]3= 1n3(n−1)(n−2)[n2ν(ν+2)(ν+1)ν−3n(n−2)(ν+1)ν2 + 2(n−1)(n−2)ν3]= 4ν3+6nν2+2n2νn3(n−1)(n−2)andµGn,ν4 =4∑i=0(4i)(−1)iGn,ν(e4−i)[Gn,ν(e1)]i= Gn,ν(e4)− 4Gn,ν(e3)Gn,ν(e1) + 6Gn,ν(e2)[Gn,ν(e1)]2− 4Gn,ν(e1)[Gn,ν(e1)]3 + [Gn,ν(e1)]4= 1n4(n−1)(n−2)(n−3)[n3(ν+3)(ν+2)(ν+1)ν−4n2(n−3)(ν+2)(ν+1)ν2+ 6n(n− 2)(n− 3)(ν + 1)ν3 − 3ν4(n− 1)(n− 2)(n− 3)]= 1n4(n−1)(n−2)(n−3)[6n3ν + 3(8 + n)n2ν2 + 6(6 + n)nν3 + 3(6 + n)ν4].The other estimates follow similarly, we thus omit the details. 3. DIRECT ESTIMATES AND ASYMPTOTIC FORMULALetBm[0,∞) = {f : |f(x)| ≤ cf (1 + xm), ∀x ∈ [0,∞)}, m > 3where cf is an absolute constant dependent on f, but independent of x. LetCm[0,∞) = C[0,∞) ∩Bm[0,∞).For each f ∈ Cm[0,∞) we consider modulus of continuity (see [14]) asΩ(f, δ) = sup|4x|<δ,x∈R+|f(x+4x)−f(x)|(1+hm)(1+xm) .Additionally, C∗m[0,∞) denotes the subspace of continuous functions f ∈Bm[0,∞), for whichlimx→∞|f(x)|(1 + xm)−1 <∞.We consider the norm by‖f‖m = sup0≤x<∞|f(x)|(1+xm) .158 Vijay Gupta and M. T. Rassias 6Theorem 6. If f ∈ C∗m [0,∞), thenlimn→∞‖(Vnf)− f‖m = 0.Proof. By the weighted Korovkin theorem due to Gadjiev [4], we know thatif f ∈ C∗m [0,∞), satisfyinglimn→∞‖(Vnei)− ei‖m = 0, i = 0, 1, 2,then we havelimn→∞‖(Vnf)(x)− f(x)‖m = 0.In order to prove the result, we use Lemma 3. Clearly, the result holds fori = 0, 1. Thus, in order to complete the proof we proceed forlimn→∞‖(Vne2)(x)− e2‖2 = limn→∞1(1+xm)[x2 + x(1+x)(2+x)n−1 − x2]= limn→∞1(1+xm)[x(1+x)(2+x)n−1]= 0.This completes the proof of the theorem. Theorem 7. For f (s) ∈ CB[0,∞) , 0 ≤ s ≤ 4, s ∈ N, x ∈ [0,∞) andn ∈ N, we have|((Vnf)− (Rnf))(x)| ≤≤ ‖fiv‖24(n−1)(n−2)(n−3)[6x+ 3(8 + n)(x2 + x(1+x)2n)+6(6 + n)(x3 + 3x2(1+x)2n +x(1+x)3(1+3x)n2)+3(6 + n)(x4 + 6x3(1+x)2n +x2(1+x)3(7+15x)n2 +x(1+x)4(1+10x+15x2)n3)]+ ‖f′′′‖3(n−1)(n−2)[2x3 + 6x2(1+x)2n +2x(1+x)3(1+3x)n2 + 3x2 + 3x(1+x)2n + x]+ ‖f′′‖2(n−1)[x2 + x(1+x)2n + x].Proof. Using Lemma 5 and Lemma 1, we haveα(x) =∞∑ν=0rn,ν(x)(µFn,ν4 + µGn,ν4 )= 124n4(n−1)(n−2)(n−3)∞∑ν=0rn,ν(x)[6n3ν + 3(8 + n)n2ν2+6(6 + n)nν3 + 3(6 + n)ν4]= 124(n−1)(n−2)(n−3)[6(Rne1)(x) + 3(8 + n)(Rne2)(x)7 Ismail-May-Baskakov operators 159+6(6 + n)(Rne3)(x) + 3(6 + n)(Rne4)(x)]= 124(n−1)(n−2)(n−3)[6x+ 3(8 + n)(x2 + x(1+x)2n)+6(6 + n)(x3 + 3x2(1+x)2n +x(1+x)3(1+3x)n2)+3(6+n)(x4+ 6x3(1+x)2n +x2(1+x)3(7+15x)n2 +x(1+x)4(1+10x+15x2)n3)].Additionally,β(x) = 13!∞∑ν=0rn,ν(x)∣∣∣µFn,ν3 − µGn,ν3 ∣∣∣= 13!∞∑ν=0rn,ν(x)4ν3+6nν2+2n2νn3(n−1)(n−2)= 13(n−1)(n−2) [2(Rne3)(x) + 3(Rne2)(x) + (Rne1)(x)]= 13(n−1)(n−2)[2x3 + 6x2(1+x)2n +2x(1+x)3(1+3x)n2 + 3x2 + 3x(1+x)2n + x].Furthermore, we haveγ(x) = 12!∞∑ν=0rn,ν(x)∣∣∣µFn,ν2 − µGn,ν2 ∣∣∣= 12!∞∑ν=0rn,ν(x) ν2+nνn2(n−1)= 12(n−1) [(Rne2)(x) + (Rne1)(x)] =12(n−1)[x2 + x(1+x)2n + x].Finally, we getδ2(x) =∞∑ν=0rn,ν(x)(bFn,ν − bGn,ν )2 = 0.Following [5, Theorem 3] and [1, Theorem 4], we derive that|((Vnf)− (Rnf))(x)| ≤ ‖f iv‖α(x) + ‖f ′′′‖β(x) + ‖f ′′‖γ(x)+2ω(f, δ(x)).Substituting the values of the above estimates, we get the desired result. Theorem 8. Let f ∈ C[0,∞) with |f(t)| ≤ C(1+ t)γ for some γ > 0, t ≥ 0.If f (q+2) exists at a point x ∈ (0,∞), then we havelimn→∞n(V (q)n f)(x)− f (q)(x)) =q(q−1)(q−2)2 f(q−1)(x) + 3q(q−1)(x+1)2 f(q)(x)+ q[(q2+2)x2+3(q+1)x+2]2 f(q+1)(x)+x(1+x)(2+x)2 f(q+2)(x).160 Vijay Gupta and M. T. Rassias 8Proof. Applying Taylor’s expansion, we have From Taylor’s theorem, wehave(V (q)n f)(x) =q+2∑v=0f (v)(x)v!(∂q∂κq (Vn(e1 − xe0)v)(κ))κ=x+(∂q∂κq (Vnψ(t, x)(e1 − xe0)q+2)(κ))κ=x=q+2∑v=0f (v)(x)v!v∑j=0(vj)(−x)v−j(∂q∂κqTn,j(κ))κ=x+(∂q∂κq (Vnψ(u, x)(e1 − xe0)q+2)(κ))κ=x:= I1 + I2where the function ψ(u, x)→ 0 as u→ x. Obviously I1 is estimated asI1 = f(q−1)(x)(q−1)!(∂q∂κqTn,q−1(κ))κ=x+f(q)(x)q![q(−x)(∂q∂κqTn,q−1(κ))κ=x+(∂q∂κqTn,q(κ))κ=x]+f(q+1)(x)(q+1)![(q+1)q2 x2(∂q∂κqTn,q−1(κ))κ=x+(q + 1)(−x)(∂q∂κqTn,q(κ))κ=x+(∂q∂κqTn,q+1(κ))κ=x]+f(q+2)(x)(q+2)![(q+2)(q+1)q3! (−x)3(∂q∂κqTn,q−1(κ))κ=x+ (q+2)(q+1)2 x2(∂q∂κqTn,q(κ))κ=x+(q + 2)(−x)(∂q∂κqTn,q+1(κ))κ=x+(∂q∂κqTn,q+2(κ))κ=x].Applying Lemma 3, we haveI1 = f(q−1)(x)(q−1)![q!(q−1)(q−2)2(n−q+2)]+ f (q)(x)[1 + 3nq(q−1)(x+1)2(n−q+2)(n−q+1) +O(n−2)]+f (q+1)(x)[nq[(q2+2)x2+3(q+1)x+2]2(n−q)(n−q+2) +O(n−2)]+f(q+2)(x)(q+2)![x32(n−q+1) +3x22(n−q−1) +nx(n−q)(n−q−1)].Thuslimn→∞n((V (q)n f)(x)− f (q)(x)) =q(q−1)(q−2)2 f(q−1)(x) + 3q(q−1)(x+1)2 f(q)(x)+ q[(q2+2)x2+3(q+1)x+2]2 f(q+1)(x)9 Ismail-May-Baskakov operators 161+x(1+x)(2+x)2 f(q+2)(x) + limn→∞nI2.Next, in view of Lemma 4, we have|I2| ≤≤ ∞∑ν=1∑2i+j≤qi,j≥0ni|ν−nκ|j |φi,j,q(κ)|κq(1+κ)2q rn,ν(κ)∫ ∞0vn,ν(u)|ψ(u, x)|.|u−x|q+2duκ=x+ |ψ(0, x)(−x)q+2|(∂q∂κq rn,0(κ))κ=x:= E1 + E2.Because of the fact ψ(u, x)→ 0 as u→ x, for a given ε > 0 there will exista δ > 0 such that |ψ(u, x)| < ε whenever |u− x| < δ.Furthermore, if m ≥ {γ, q + 2}, and m a positive integer|(u− x)q+2ψ(u, x)| ≤M |u− x|mfor |u− x| ≥ δ. Thus|E1| ≤∞∑ν=1∑2i+j≤qi,j≥0ni|ν − nx|j |φi,j,q(x)|xq(1+x)2q rn,ν(x)(ε∫|u−x|<δvn,ν(u)|u− x|q+2du+M∫|u−x|≥δvn,ν(u)|u− x|mdu):= J1 + J2.Let us considerK = sup2i+j≤qi,j≥0|φi,j,q(x)|xq(1+x)2q .Using Schwarz’s inequality, Lemma 1, Lemma 2 and Lemma 3, we deducethatJ1 = ε K∞∑ν=1∑2i+j≤qi,j≥0ni|ν − nx|jrn,ν(x)(∫ ∞0vn,ν(u)du)1/2(∫ ∞0vn,ν(u)(e1 − xe0)2q+4du)1/2≤ ε K∑2i+j≤qi,j≥0ni+j( ∞∑ν=0rn,ν(x)(νn − x)2j− x2jrn,0(x))1/2(Vn((e1 − xe0)2q+4, x)− x2q+4rn,0(x))1/2162 Vijay Gupta and M. T. Rassias 10= ε∑2i+j≤qi,j≥0ni+j{Ox(1nj)+Ox(1ns)}1/2{Ox(1nq+2)+Ox(1np)}1/2,for any s, p > 0, choosing s and p such that s > j and p > q + 2, we obtainJ1 ≤ ε∑2i+j≤qi,j≥0ni+jOx(1nj/2)Ox(1nq/2+1)= ε.Ox(n−1).Since ε > 0 is arbitrary, nJ1 → 0 as n→∞.Applying again Schwarz’s inequality, Lemma 1 and Lemma 3, we obtainJ2 ≤ M1∑2i+j≤qi,j≥0ni+j( ∞∑ν=0(νn − x)2jrn,ν(x)− x2jrn,0(x))1/2( ∞∑ν=1rn,ν(x)∫|u−x|≥δvn,ν(u)(e1 − xe0)2mdu)1/2≤ M1∑2i+j≤qi,j≥0ni+j{Ox(1nj)+Ox(1ns)}1/2{Ox(1nm)+Ox(1nl)}1/2,for any s, l > 0. We now choose s, l such that s > j, l > m. Then, we derivethatJ2 ≤ M1∑2i+j≤qi,j≥0ni+jOx(1nj/2)Ox(1nm/2)= Ox(1n(m−q)/2),which implies that nJ2 → 0, as n→∞, on choosing m > q + 2.From the above estimates of J1 and J2, nE1 → 0, as n → ∞. Finally, wehave|E2| = |ψ(0, x)(−x)q|(∂q∂κq rn,0(κ))κ=x.Since |ψ(0, x)(−x)q| < N1 for some N1 > 0, also(∂q∂κq rn,0(κ))κ=x=[∂q∂κq(e−nκ/(1+κ))]κ=xwhich yields that nE2 → 0 as n → ∞. By collecting the estimates of E1 andE2, we obtain nI2 → 0 as n→∞. Therefore, combining I1 and I2, the requiredresult follows. Corollary 9. If f ∈ C[0,∞) and |f(t)| ≤ C(1 + t)γ for some γ > 0, t ≥ 0with the existence of f ′′ in x ∈ (0,∞), then we havelimn→∞n((Vnf)(x)− f(x)) = x(x2+3x+2)2 f′′(x).11 Ismail-May-Baskakov operators 163REFERENCES[1] A. M. Acu and I. Raşa, New estimates for the differences of positive linear operators,Numer. Algor., 73 (2016), pp. 775–789, https://doi.org/10.1007/s11075-016-0117-8.[2] O. Agratini, Approximation properties of a family of integral type operators, Positivity25 (2021) no. 1, pp. 97–108, doi.org.10.1007/s11117-020-00752-y.[3] P. N. Agrawal and K. J. Thamer, Approximation of unbounded functions by a newsequence of linear positive operators, J. Math. Anal. Appl., 225 (1998), pp. 660–672,https:doi.org.10.1006/jmaa.1998.6036.[4] A. D. Gadjiev, Theorems of Korovkin type, Math. Zametki, 20 (1976) no. 5, pp. 781–786.[5] V. Gupta, General estimates for the difference of operators, Comp. Math. Methods, 1(2019):e1018, pp. 1–12, https://doi.org.doi.org/10.1002/cmm4.1018.[6] V. Gupta and G. Agrawal, Approximation for link Ismail-May operators, Ann. Funct.Anal., 11 (2020), pp. 728–747, https://doi.org.10.1007/s43034-019-00051-y.[7] V. Gupta and G. C. Greubel, Moment estimations of new Szász–Mirakyan–Durrmeyeroperators, Appl. Math. Comput., 271 (2015), pp. 540–547, https://doi.org.10.1016/j.amc.2015.09.037.[8] V. Gupta and M. A. Noor, Convergence of derivatives for certain mixed Szász–Betaoperators, J. Math. Anal. Appl., 321 (2006) no. 1, pp. 1–9, https://doi.org.10.1016/j.jmaa.2005.07.036.[9] V. Gupta and Th. M. Rassias, Direct estimates for certain Szász type operators, Appl.Math. Comput., 251 (2015), pp. 469–474, https://doi.org.10.1016/j.amc.2014.11.078.[10] V. Gupta and M. Th. Rassias, Moments of Linear Positive Operators and Approxi-mation, Series: SpringerBriefs in Mathematics, Springer Nature Switzerland AG, 2019,https://doi.org.10.1007/978-3-030-19455-0.[11] V. Gupta, D. K. Verma and P. N. Agrawal, Simultaneous approximation by certainBaskakov–Durrmeyer–Stancu operators, J. Egyptian Math. Soc., 20 (2012) no. 3, pp.183–187, https://doi.org.10.1016/j.joems.2012.07.001.[12] M. Ismail, Polynomials of binomial type and approximation theory, J. Approx. Theory,23 (1978), pp. 177–186, https:doi.org.10.1016/0021-9045(78)90105-3.[13] M. Ismail and C. P. May, On a family of approximation operators, J. Math. Anal.Appl., 63 (1978), pp. 446–462, https://doi.org.10.1016/0022-247X(78)90090-2.[14] N. Ispir, On modified Baskakov operators on weighted spaces, Turk. J. Math., 26 (2001)no. 3, pp. 355–365.Received by the editors: June 30, 2021; accepted: August 29, 2021; published online: Feb-ruary 17, 2022.

Asymptotic formula in simultaneous approximation for certain Ismail-May-Baskakov operators

https://ictp.acad.ro/jnaat/journal/article/download/1235/1208

Asymptotic formula in simultaneous approximation for certain Ismail-May-Baskakov operators

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals

Directory of Open Access Journals