Rate of convergence for certain families of summation integral type operators

J Math Anal Appl 296 24 68 618 wwwelseviercom/locate/jmaa Rate of convergence for certain families of summation integral type operators Vijay Gupta a,,mkgupta b a School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 1145, India b Department of Mathematics, Ch Charan Singh University, Meerut 254, India Received 14 August 23 Available online 8 July 24 Submitted by HM Srivastava Abstract In this paper, we study the rate of convergence for the iterative combinations of a certain family of linear positive operators in terms of higher-order integral modulus of smoothness We have used the technique of linear approimating method, namely Steklov mean, to prove the main result In the end, we propose some other sequences of linear positive operators and obtain the recurrence formulae for the central moments, Voronovskaja type asymptotic formulae and error estimations 24 Elsevier Inc All rights reserved Keywords: Linear positive operators; Iterative combinations; Steklov mean; Modulus of smoothness 1 Introduction Very recently, Srivastava and Gupta [8] defined a certain family of linear positive operators, which include some well-known operators as special cases, they estimated the rate of convergence for functions of bounded variation For f L p [,, p 1, the mied summation integral type operators see, eg, [2,5] discussed in this paper are defined as * Corresponding author E-mail addresses: vijay@nsitacin V Gupta, mkgupta22@hotmailcom MK Gupta 22-247X/$ see front matter 24 Elsevier Inc All rights reserved doi:1116/jjmaa24429

where V n f, = V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 69 = n K n, t = n K n, tf t dt b n,v s n,v 1 tf t dt 1 n f, 1 b n,v s n,v 1 t 1 n δt, δt being Dirac delta function, and n v 1 b n,v = v 1 n v nt ntv, s v n,v t = e v! It has been easily verified that the operators V n are linear positive operators It turns out that the order of approimation by these operators is at best On 1, n,howsoever smooth the function may be Thus to improve the order of approimation, we may consider some combinations of the operators 1 In this contet, the linear combinations of the operators V n were recently considered in [2] for improving the order of approimation Yet another approach for improving the order of approimation is the iterative combinations due to Micchelli [7], who improved the order of approimation of Bernstein polynomials; the results of [7] were later sharpened in [3] One cannot apply the iterative combinations easily for all linear positive operators; for eample, in the sequence of operators see [1,6,8] t is not mapped eactly to, so it is much more difficult to handle the iterative combinations for the operators discussed in these papers But for the operators 1, we have V n t, =, which is the interesting property of these operators V n, thus one can consider the iterative combinations of 1 to improve the rate of convergence Here we consider the iterative combinations V n,k f, of the operators V n,whichforf L p [, are defined as V n,k f, = [ I I V n k] ft, k = 1 r1 k V r r n ft,, 2 where V r n denotes the rth iterate superposition of the operator V n It is obvious that Bernstein polynomials summation type operators are as such not L p approimation methods Nevertheless several linear positive operators of summation type have been appropriately modified to become L p -approimation methods The operators studied in the present paper make it possible to approimate pth 1 p< power Lebesgue integrable functions on [, This, along with the application of iterative combinations V n,k of the operators V n, motivated us to study in this direction Several other approimation properties on some summation integral type operators were studied in [1,5,6], etc In the present paper, we consider iterative combination due to Micchelli for the operators V n defined by 2 and obtain an error estimate in terms of higher-order integral modulus r=1

61 V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 of smoothness in L p -norm In the end, we propose some other mied summation integral type operators and give their central moments in the form of recurrence relations 2 Auiliary results In this section we mention some basic lemmas Lemm Let the function µ n,m, m N the set of non-negative integers be defined by µ n,m = n b n,v s n,v 1 tt m dt m 1 n, < Then 2 µ n, = 1, µ n,1 =, µ n,2 = n Furthermore, the following recurrence relation holds: nµ n,m1 = 1 µ 1 n,m m2 µ n,m 1 mµ n,m, m 1 It follows from the above recurrence relation that µ n,m = O n [m1/2] n, < ; m N Remark 1 For each m N the mth order moment µ {q} n,m for the operators Vn q is defined by µ {q} n,m = Vn q t m, 3 For q = 1, µ 1 n,m µ n,m Lemm There holds the following relation: m m j µ n,m {q1} m 1 = j i! Di µ {q} n,m j µ n,ij, j= i= D d d Lemma 3 We have µ {q} n,m = O n [m1/2] n, < ; m N Proof The proof of the above lemma easily follows by using the principle of mathematical induction and Lemm By direct application of Lemmas 2 and 3, we have V n,k t i, = O n k n, 4

V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 611 where V n,k is the iterative combination defined by 2 Throughout the present paper, we consider < < <a 3 <b 3 <b 2 <b 1 < and I r =[a r,b r ], r = 113 Lemma 4 Let f L p [,, p>1,alsoletf 2k eists on I 1 with f 2k 1 ACI 1 and f 2k L p I 1, then for n sufficiently large, we have Vn,k f, f Lp I 2 C 1n k{ f 2k Lp I 2 f L p [, } 5 Also, if f L 1 [,, f 2k 1 L 1 I 1 with f 2k 2 ACI 1 and f 2k 1 BVI 1, then for n sufficiently large, we have V n,k f, f L1 I 2 C 2n k{ f 2k 1 BVI1 f 2k 1 L1 I 2 f L1 [, }, 6 where the constants C 1 and C 2 are independent of f and n Proof First let p>1 By the hypothesis, for all t [, and I 2,wehave 2k 1 V n,k f, f= i=1 f i V n,k t i, i! 1 2k 1! V n,k φt t t w 2k 1 f 2k w dw, V n,k Ft, 1 φt, = E 1 E 2 E 3, where φt denotes the characteristic function of I 1 and 2k 1 t i Ft,= ft f i i! i= Applying 4 and the interpolation property due to Goldberg and Meir [4], we obtain 2k 1 E 1 Lp I 2 C 3 n k f i Lp I 2 C 4n k{ f Lp I 2 f 2k Lp I 2 } i=1 To estimate E 2,letH f be the Hardy Littlewood maimal function [9] of f 2k on I 1 Using Holder s inequality and Lemm, we have E 2 Vn φt t t w 2k 1 f 2k w dw, { V n φt t 2kq, } 1/q { V n φt Hf t p, } 1/p b1 C 5 n k K n, t hf t p dt 1/p

612 V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 Therefore, we have E 2 Lp I 2 C 6 n k f 2k Lp I 1 For t [, \[,b 1 ], I 2, there eists δ> such that t δ Thus E3 V n Ft, 1 φt, δ 2k V n Ft, t 2k, δ [V 2k n ft t 2k, 2k 1 = E31 E 32 Applying Holder s inequality and Lemm, we have δ 2k{ V n ft p, } 1/p{ Vn t 2kq, } 1/q E 31 C 7 n k{ V n ft p } 1/p, Finally, by Fubini s theorem, we obtain E 31 p L p I 2 = b2 i= E p 31 d C8 n pk f p L p [,, also by the interpolation property [4], we have E 32 Lp I 2 C 9n k{ f Lp I 2 f 2k Lp I 2 } Thus E 3 Lp I 2 C 1 n k{ f Lp [, f 2k Lp I 2 } f i V n t 2ki, ] i! Combining E 1, E 2 and E 3, we get 5 Net suppose p = 1 By the hypothesis, for almost all I 2 and for all t [,,we have 2k 1 V n,k f, f= i=1 f i V n,k t i, i! 1 2k 1! V n,k φt t t w 2k 1 f 2k 1 w dw, V n,k Ft, 1 φt, = H 1 H 2 H 3 Applying Lemm and the interpolation property due to Goldberg and Meir [4], we have H 1 L1 I 2 C 11 n k{ f L1 I 2 f 2k 1 L1 I 2 } Net, for the estimation of H 2 we proceed as follows:

V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 613 H 2 V n φt b 2 b 1 t t w 2k 1 f 2k 1 w dw, K n, t t 2k 1 t L1 I 2 df 2k 1 w dt d For each n there eists a nonnegative integer r = rn satisfying rn 1/2 < mab 1, b 2 r 1n 1/2 H 2 r l= b 2 { l1n 1/2 ln 1/2 φtk n, t t l1n 1/2 r l= b 2 { ln 1/2 2k 1 φw df 2k 1 w } dt d l1n 1/2 φtk n, t t l1n 1/2 φt 2k 1 df 2k 1 w } dt d Suppose φ w,c,d denotes the characteristic function of the interval [ cn 1/2, dn 1/2 ],wherec,d are positive integers Then we have { r H2 l 4 n 2 l=1 b1 b 2 l1n 1/2 φ w,,l 1 { r l 4 n 2 l=1 b1 ln 1/2 φtk n, t t b 2 ln 1/2 φ w, l 1, df 2k 1 w } dt 2k3 l1n 1/2 φtk n, t t df 2k 1 w } dt 2k3

614 V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 b 2 { a1 n 1/2 2k 1 φtk n, t t n 1/2 b1 φ w, 1, 1 df 2k 1 w } dt d Applying Lemm and using Fubini s theorem, we obtain { r b1 w H2 C 12n 2k1/2 l 4 df 2k 1 w l=1 b 1 wl1n 1/2 w w l1n 1/2 d df d 2k 1 w b 1 wn 1/2 w n 1/2 d df 2k 1 w } C 13 n k f 2k 1 w BVI1 Thus H 2 L1 I 2 C 14 n k f 2k 1 BVI1 Finally we estimate H 3 ;forallt [, \[,b 1 ] and all [,b 2 ], we choose δ> such that t δ Thus H L1 3 I 2 Vn Ft, 1 φt, L1 I 2 b 2 2k 1 r= K n, t ft 1 φt dt d 1 r! b 2 K n, t f r t t r 1 φt dt d Applying Lemm and the interpolation property due to Goldberg and Meir [4], the second term in the right-hand side of the above epression is bounded by C 15 n k { f L1 [, f 2k 1 L1 I 2 } The first term is estimated as follows For t sufficiently large there eist positive constants C 16 and C 17 satisfying the condition t 2k /t 2k 1>C 17,forallt C 16, t [,b 2 ] Therefore, using Fubini s theorem and Lemm, we have b 2 K n, t ft 1 φt dt d C 16 b 2 b 2 C 16 K n, t ft 1 φt dt d

V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 615 C 18 n k C 16 ft dt 1 C 19 n k f L1 [, C 17 C 16 b 2 t 2k K n, t ft ddt t 2k 1 Collecting the estimates of H 1, H 2 and H 3, the required estimate 6 follows 3 Error estimation We first define the linear approimating method viz Steklov mean which is the main tool to prove the error estimation Let f L p [a,b],1 p<, then for sufficiently small η> the Steklov mean f η,m of mth order corresponding to f is defined by f η,m = η m η/2 η/2 η/2 η/2 { ft 1 m 1 m u ft} dt 1 dt 2 dt m, where u = m i=1 t i, t [a,b], and m η ft is the mth forward differences with step length η It is verified that f η,m has derivatives with respect to order m, f η,m n 1 AC[,b 1 ] and mth derivatives of f η,m eists almost everywhere and belonging to L p [,b 1 ]Moreover, i f r η,m Lp [,b 2 ] K 1 η r ω r f,η,p,[,b 1 ], r = 1, 2,,m; ii f f η,m Lp [,b 2 ] K 2 ω m f,η,p,[,b 1 ]; iii f η,m Lp [,b 2 ] K 3 f Lp [,b 1 ]; iv f m η,m Lp [,b 2 ] K 4 η m f Lp [,b 1 ], where K i s i = 1, 2, 3, 4 are certain constants independent of f and η Theorem 1 Let f L p [,, p 1 Then for n sufficiently large, we have Vn,k f, f Lp I 2 C { 2 ω2k f,n 1/2,p,I 1 n k } f Lp [,, where C 2 is a constant independent of f and n Proof By linearity property, we have Vn,k f, f Lp I 2 Vn,k f f η,2k, Lp I 2 Vn,k f η,2k, f η,2k Lp I 2 f η,2k f Lp I 2 = 1 2 3 First we estimate 1,letχt be the characteristic function of the interval I 3 Thus

616 V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 V n f fη,2k t, = V n χtf fη,2k t, V n 1 χt f fη,2k t, Now applying Fubini s theorem, we have b 2 Vn χt f fη,2k t, p d b 3 b 2 K n, t f f η,2k p ddt a 3 f f η,2k p L p I 3 Net using Lemm and Holder s inequality, for allp 1, we have V n 1 χt f fη,2k t, Lp I 2 C 21n k f f η,2k Lp [, Thus 1 C 22 { ω2k f,η,p,i 1 n k f Lp [, } Also it is observed that f 2k 1 η,2k BVI3 = f 2k η,2k L 1 I 3 Applying Lemma 4 and the property of Steklov mean s, we get 2 C 23 n k{ 2k f Lp I 3 f } η,2k Lp {, η,2k C 24 n k{ η 2k ω 2k f,η,p,i 1 f Lp [, } Finally, by the property of Steklov mean s, we get 3 f η,2k f Lp I 2 C 25 ω 2k f,η,p,i 1 Choosing η = n 1/2, and combining the estimates of 1, 2 and 3, the theorem follows 4 Some eamples Let s n,v, b n,v and p n,v be Szasz, Baskakov and Beta basis functions, respectively, which are defined as n nv n v 1 s n,v = e, b n,v = v 1 n v, v! v p n,v = 1 Bn,v 1 v 1 n v 1 To approimate Lebesgue integrable functions on the interval [,, we propose some other new summation integral type operators

V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 617 Szasz Baskakov type operators: S n f, = n 1 s n,v b n,v 1 tf t dt e n f If S n,m S n t m,, then they satisfy the recurrence relation: n m 2S n,m1 = [ S n,m 1 m 2S n,m 1 ] [ m 2m 1 ] S n,m ; n>m 2 Beta Szasz type operators: P n f, = p n,v s n,v 1 tf t dt 1 n 1 f If P n,m P n t m,, we have the recurrence relation: np n,m1 = 1 [ P 1 n,m mp n,m 1 ] m P n,m mp n,m 1 Szasz Beta type operators: B n f, = s n,v p n,v 1 tf t dt e n f If B n,m B n t m,, we have the recurrence relation: n m 1B n,m1 = [ B n,m 1 m 2B n,m 1 ] [ m 1 2m ] B n,m ; n>m 1 From the above recurrence relations for different operators, we obtain the following Voronovskaja type asymptotic formulae and error estimations Theorem 2 Let f be bounded and integrable on [, and has second derivative at a point [,,then i ii iii iv lim n[ V n f, f ] = n 2 f 2 ; 2 lim n[ S n f, f ] = 2f 1 n lim n n[ P n f, f ] = f 1 lim n[ B n f, f ] = f 1 n 2 f 2 ; 2 2 f 2 ; 2 2 f 2 2

618 V Gupta, MK Gupta / J Math Anal Appl 296 24 68 618 Theorem 3 Let f C[, Then for [,a], <a<, we have i V n f, f C[,a] [ 1 aa 2 ] ω f,n 1/2 ; ii Sn f, f [ C[,a] 1 n[naa 2 ] 6a2 ] ω f,n 1/2 ; n 2n 3 iii Pn f, f [ ] a1 an 2 na C[,a] 1 ω f,n 1/2 ; n iv B n f, f [ ] C[,a] 1 n[a2 n 2 2na] ω f,n 1/2, n 1n 2 where ωf, is the modulus of continuity of f and is the sup norm on the interval [,a] Remark 2 It is observed that the mied summation integral type operators are easier to construct but each of these operators have different properties in approimation theory It may be remarked here that the above mentioned operators S n, P n and B n are linear positive operators, but we cannot apply the iterative combinations for these operators the reason is mentioned in the introduction Therefore, for these operators we cannot obtain the result analogous to our Theorem 1 Also as another eample we cannot obtain the rate of convergence for bounded variation functions easily for the above mentioned operators analogous to the results of [6,8] The main problem is that one cannot find a relation between summation and integration having different basis functions These problems are still unresolved and these may be treated as open problems for the readers Acknowledgment The authors are grateful to the reviewers for the critical review, leading to a better presentation of the paper References [1] PN Agrawal, KJ Thamer, Approimation of unbounded functions by a new sequence of linear positive operators, J Math Anal Appl 225 1998 66 672 [2] PN Agrawal, AJ Mohammad, On L p approimation by a linear combination of a new sequence of linear positive operators, Turkish J Math 27 23 389 45 [3] PN Agrawal, HS Kasana, On the iterative combinations of Bernstein polynomials, Demonstratio Math 27 1984 777 783 [4] S Goldberg, V Meir, Minimum moduli of ordinary differential operators, Proc London Math Soc 23 1971 1 15 [5] MK Gupta, V Vasishtha, On the iterative combinations of a sequence of linear positive operators, Math Comput Modelling 39 24 521 527 [6] V Gupta, Rate of convergence by a new sequence of linear positive operators, Comput Math Appl 45 23 1895 194 [7] CA Micchelli, The saturation class and iterates of the Bernstein polynomials, J Appro Theory 8 1973 1 18 [8] HM Srivastava, V Gupta, A certain family of summation integral type operators, Math Comput Modelling 37 23 137 1315 [9] AF Timan, Theory of Approimation of Functions of a Real Variable, Hindustan, Delhi, 1966