This note is motivated by the results on the strong approximation of 2&#928;-periodic functions by means of trigonometric Fourier series.In this note is investigated certain class of positive linear operators in the polynomial weighted spaces. We introduce the strong differences of functions and their operators and we give the Jackson type theorems for them. We give also some corollaries.</p

Rempulska, Lucyna

Skorupka, Mariola

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On Strong Approximation of Functions by Certain Linear Operators

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Mathematical Journal of OkayamaUniversityVolume 46, Issue 1 2004 Article 10JANUARY 2004On Strong Approximation of Functions byCertain Linear OperatorsLucyna Rempulska∗ Mariola Skorupka†∗University of Technology Piotrowo†University of Technology PiotrowoCopyright c©2004 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouOn Strong Approximation of Functions byCertain Linear OperatorsLucyna Rempulska and Mariola SkorupkaAbstractThis note is motivated by the results on the strong approximation of 2Π-periodic functionsby means of trigonometric Fourier series.In this note is investigated certain cla of positive linearoperators in the polynomial weighted spaces. We introduce the strong differences of functions andtheir operators and we give the Jackson type theorems for them. We give also some corollaries.KEYWORDS: linear operator, degree of approximation, strong, approximation.Math. J. Okayama Univ. 46 (2004), 153{161ON STRONG APPROXIMATION OF FUNCTIONSBY CERTAIN LINEAR OPERATORSLucyna REMPULSKA and Mariola SKORUPKAAbstract. This note is motivated by the results on the strong approxi-mation of 2¼-periodic functions by means of trigonometric Fourier series.In this note is investigated certain class of positive linear operators inthe polynomial weighted spaces. We introduce the strong di®erences offunctions and their operators and we give the Jackson type theorems forthem. We give also some corollaries.1. Introduction1.1. The problem of strong approximation of functions connected withFourier series was examined in many papers presented by G. Alexits, K.Tandori, L. Leindler, R. Taberski, V. Totik and other authors (see [5]).The monograph [5] is devoted to the strong approximation of 2¼-periodicfunctions belonging to various classes by the means of trigonometric Fourierseries.For example, if Sn(f ;x) is the n-th partial sum of trigonometric Fourierseries of f , then the n-th (C; 1)-mean of this series is de¯ned by the formula¾n(f ;x) :=1n+ 1nXk=0Sk(f ;x); n 2 N0 = f0; 1; :::g :The n-th strong (C; 1)-mean of this series is de¯ned as followsHqn(f ;x) :=(1n+ 1nXk=0jSk(f ;x)¡ f(x)jq) 1q; n 2 N0;where q is a ¯xed positive number. It is clear thatj¾n(f ;x)¡ f(x)j · H1n(f ;x)andHqn(f ;x) · Hpn(f ;x); 0 < q < p <1;for all x 2 R and n 2 N0. The last inequalities show that examination ofthe strong means of Fourier series is useful.The purpose of this note is to show that investigation of the strong ap-proximation connected with linear operators is also useful.Mathematics Subject Classi¯cation. 41A25.Key words and phrases. linear operator, degree of approximation, strongapproximation.1531Rempulska and Skorupka: On Strong Approximation of Functions by Certain Linear OperatorsProduced by The Berkeley Electronic Press, 2004154 L. REMPULSKA AND M. SKORUPKA1.2. In [2] were examined approximation properties of the Sz¶asz-Mirakjanoperators ([6])(1) Sn(f ;x) := e¡nx1Xk=0(nx)kk!fµkn¶;and the Baskakov operators ([1])(2) Vn(f ;x) :=1Xk=0µn¡ 1 + kk¶xk(1 + x)¡n¡k fµkn¶;n 2 N = f1; 2; :::g, x 2 R0 = [0;1), for the functions f belonging to thepolynomial weighted spaces Cp, p 2 N0. The space Cp, p 2 N0, is associatedwith the weighted function(3) w0(x) := 1; wp(x) := (1 + xp)¡1 if p ¸ 1;and it is the set of all real-valued functions f for which wpf is uniformlycontinuous and bounded on R0 and the norm is de¯ned by the formula(4) kfkp ´ kf( ¢ )kp := supx2R0wp(x) jf(x)j:The author proved in [2] that for every p 2 N0 there exists a positiveconstant M(p) depending only on p such that for every f 2 Cp there holds(5) wp(x) jVn(f ;x)¡f(x)j ·M(p)!2³f ;p(x+ x2)=n´n 2 N; x 2 R0;where !2(f ; ¢) is the second modulus of smoothness of f . From (5) it followsthat(6) limn!1Vn(f ;x) = f(x); x 2 R0; f 2 Cp;and this convergence is uniform on every interval [x1; x2], x1 ¸ 0:The analogous results for the Sz¶asz-Mirakyan operators are given in [2]also.In this note we introduce certain class of linear operators in the spacesCp and we de¯ne the strong di®erences for them. We give two theorems andsome corollaries on these strong di®erences.We shall denote by Mk(®; ¯), k 2 N , suitable positive constants depend-ing only on indicated parameters ®, ¯:2. Definitions and preliminary results2.1. Let ­ be the set of all in¯nite matrices A = [ank], n 2 N , k 2 N0, offunctions in C0 having the following properties:(i) ank(x) ¸ 0 for x 2 R0, n 2 N , k 2 N0,(ii)P1k=0 ank(x) = 1 for x 2 R0, n 2 N ,2Mathematical Journal of Okayama University, Vol. 46 [2004], Iss. 1, Art. 10http://escholarship.lib.okayama-u.ac.jp/mjou/vol46/iss1/10ON STRONG APPROXIMATION OF FUNCTIONS 155(iii) for every n; r 2 N the series P1k=0 krank(x) is uniformly convergenton R0 and its sum is a function belonging to the space Cr,(iv) for every r 2 N there exists positive constant M1(r;A) independenton x 2 R0 and n 2 N such that for the functions(7) Tn;2r(x;A) :=1Xk=0ank(x)µkn¡ x¶2r; x 2 R0;(belonging to C2r) there holdskTn;2r( ¢ ;A)k2r ·M1(r;A)n¡r; n 2 N:Choosing A 2 ­ and p 2 N0 we de¯ne for f 2 Cp the following positivelinear operators(8) Ln(f ;A;x) :=1Xk=0ank(x) fµkn¶; n 2 N; x 2 R0:The properties (i)-(iv) of the matrix A imply that the operators Ln(f ;A)are well-de¯ned and(9) Ln(1;A;x) = 1 for x 2 R0; n 2 N;and by (8) and (9) we have(10) Ln(f ;A;x)¡ f(x) =1Xk=0ank(x)µfµkn¶¡ f(x)¶:For Ln(f ;A) and f 2 Cp we de¯ne the strong di®erence with the powerq > 0 as follows:(11) Hqn(f ;A;x) :=( 1Xk=0ank(x)¯¯¯¯fµkn¶¡ f(x)¯¯¯¯q) 1q; x 2 R0; n 2 N:Then we see that by the properties (i)-(iv) of A the Hqn(f ;A) are well-de¯ned for every f 2 Cp, p 2 N0, and q > 0: Moreover (10) and (11) implythatHqn(f ;A;x) = fLn (jf(t)¡ f(x)jq;A;x)g1q ;(12)jLn(f ;A;x)¡ f(x)j · H1n(f ;A;x);(13)and by the HÄolder inequality and (12) and (9)(14) Hqn(f ;A;x) · Hrn(f ;A;x); 0 < q < r <1;for every f 2 Cp, x 2 R0 and n 2 N:3Rempulska and Skorupka: On Strong Approximation of Functions by Certain Linear OperatorsProduced by The Berkeley Electronic Press, 2004156 L. REMPULSKA AND M. SKORUPKA2.2. First we shall give some properties of the operators Ln(f ;A):Lemma 2.1. Let A 2 ­, p 2 N0 and q > 0 be ¯xed. Then there existsM2 ´M2(p; q; A) > 0 such that(15) k ¡Ln ¡(wp(t))¡q;A; ¢ ¢¢ 1q kp ·M2; n 2 N;and for every f 2 Cp(16) k (Ln (jf jq;A; ¢ ))1q kp ·M2 kfkp; n 2 N:Proof. a) Let q = 1. From (3), (8) and (9) we getLn(1=wp(t);A;x) = 1 + Ln (tp;A;x)· 1 + 2p (Ln (jt¡ xjp;A;x) + xp)· 2p ¡(wp(x))¡1 + Ln (jt¡ xjp;A;x)¢ ; x 2 R0; n 2 N:By the HÄolder inequality and (9), we haveLn (jt¡ xjp;A;x) ·¡Ln¡(t¡ x)2p;A;x¢¢ 12and by the inequality (wp(x))2 · w2p(x) for x 2 R0, we getwp(x)Ln(1=wp(t);A;x) · 2p³1 +¡w2p(x)Ln¡(t¡ x)2p;A;x¢¢ 12´ :Applying (7) and the inequality given in (iv), we obtainwp(x)Ln(1=wp(t);A;x) ·M2(p;A) for x 2 R0; n 2 N;which by (4) implies (15).b) Let q ¸ 2 be integer. From (3) we get the following inequalities(17) (wp(x))q · wpq(x); (wp(x))¡q · 2q(wpq(x))¡1;for x 2 R0. Applying (17) we can writewp(x)¡Ln¡(wp(t))¡q;A;x¢¢ 1q · 2 (wpq(x)Ln (1=wpq(t);A;x))1q· 2 (kLn (1=wpq(t);A; ¢ ) kpq)1q ;and we can apply (15) for the last norm. This implies (15).c) Let 0 < q 62 N . Then by the HÄolder inequality and (9) we get¡Ln¡(wp(t))¡q;A;x¢¢ 1q · ¡Ln ¡(wp(t))¡r;A;x¢¢ 1r ; x 2 R0;for every 0 < q < r < 1. In particular setting r = [q] + 1 ([q] denotes theintegral part of q), we have°°°¡Ln ¡(wp(t))¡q;A; ¢ ¢¢ 1q °°°p·°°°¡Ln ¡(wp(t))¡r;A; ¢ ¢¢ 1r °°°p;4Mathematical Journal of Okayama University, Vol. 46 [2004], Iss. 1, Art. 10http://escholarship.lib.okayama-u.ac.jp/mjou/vol46/iss1/10ON STRONG APPROXIMATION OF FUNCTIONS 157and by the case b) we obtain (15) for 0 < q 62 N: Thus the proof of (15) iscompleted.If f 2 Cp and q > 0, then by (8) and (4) we get°°°(Ln (jf jq ;A; ¢ )) 1q °°°p· kfkp°°°¡Ln ¡(wp(t))¡q;A; ¢ ¢¢ 1q °°°p;and by (15) we obtain (16). ¤Lemma 2.2. Let A, p and q be as in Lemma 2.1. Then there exists M3 ´M3(p; q; A) > 0 such that(18) wp(x)½Lnµµ jt¡ xjwp(t)¶q;A;x¶¾ 1q·M3¡Ln¡(t¡ x)2s;A;x¢¢ 12sfor all x 2 R0 and n 2 N , where(19) s =(q if q 2 N;[q] + 1 if 0 < q 62 N:Proof. By (8) and by the HÄolder inequality we getwp(x)µLnµµ jt¡ xjwp(t)¶q;A;x¶¶ 1q· wp(x)¡Ln¡(wp(t))¡2q;A;x¢¢ 12q ££ ¡Ln ¡(t¡ x)2q;A;x¢¢ 12qfor all x 2 R0, n 2 N . Applying (15) and the inequality(20) (Ln (jt¡ xjq;A;x))1q · (Ln (jt¡ xjr;A;x))1r ; x 2 R0; n 2 N;for 0 < q < r <1, we easily obtain the desired estimation (18). ¤Lemma 2.2 and the property (iv) of A imply the followingCorollary 1. For every matrix A 2 ­, p 2 N0 and q > 0 there existsM4 ´M4(p; q; A) > 0 such thatwp(x)½Lnµµ jt¡ xjwp(t)¶q;A;x¶¾ 1q·M4 1 + xpnfor all x 2 R0 and n 2 N .3. Theorems and corollaries3.1. First we shall give two theorems on the strong di®erences Hqn(f ;A)de¯ned by (11). We shall use the modulus of continuity of f 2 Cp ([3])(21) !(f ; t) = sup0·h·tk¢hf( ¢ )kp; t ¸ 0;where ¢hf(x) = f(x+ h)¡ f(x).5Rempulska and Skorupka: On Strong Approximation of Functions by Certain Linear OperatorsProduced by The Berkeley Electronic Press, 2004158 L. REMPULSKA AND M. SKORUPKAIt is known ([3]) that limt!0+ !(f ; t) = 0 for every f 2 Cp, p 2 N0:Let C1p be the class of all f 2 Cp having the ¯rst derivative on R0 andf 0 2 Cp:Theorem 3.1. Suppose that A 2 ­, p 2 N0 and q > 0. Then there existM5 ´M5(p; q; A) > 0 such that for every f 2 C1p there holds(22) wp(x)Hqn(f ;A;x) ·M5 kf 0kp (Tn;2s(x;A))12s ;for all x 2 R0 and n 2 N , where Tn;2s( ¢ ;A) is de¯ned by (7) and s is givenby (19).Proof. For f 2 C1p and t; x 2 R0 we havejf(t)¡ f(x)j =¯¯¯¯Z txf 0(u)du¯¯¯¯· kf 0kpµ1wp(t)+1wp(x)¶jt¡ xj:From this we getHqn(f ;A;x) · kf 0kpµLnµµ1wp(t)+1wp(x)¶qjt¡ xjq;A;x¶¶ 1qand furtherwp(x)Hqn(f ;A;x) · 2 kf 0kp(wp(x)µLnµµ jt¡ xjwp(t)¶q;A;x¶¶ 1q++ (Ln (jt¡ xjq;A;x))1qofor x 2 R0 and n 2 N: Appling Lemma 2.2 and (7) and the inequality (20)with r = 2q, we obtainwp(x)Hqn(f ;A;x) · 2°°f 0°°p(Tn;2s(x;A))12s (M3(p; q; A) + 1)for x 2 R0, n 2 N and s de¯ned by (19). Thus the proof of (22) is completed.¤Theorem 3.2. Let A 2 ­, p 2 N0 and q > 0. Then there exists M6 ´M6(p; q; A) = const: > 0 such that for every f 2 Cp we have(23) wp(x)Hqn(f ;A;x) ·M6 !µf ;1 + xpn¶;for all x 2 R0 and n 2 N , where !(f ; ¢ ) is the modulus of continuity of f ,de¯ned by (21).Proof. Let q ¸ 1. We shall apply the Stieklov function fh for f 2 Cp:fh(x) :=1hZ h0f(x+ u)du; x 2 R0; h > 0:6Mathematical Journal of Okayama University, Vol. 46 [2004], Iss. 1, Art. 10http://escholarship.lib.okayama-u.ac.jp/mjou/vol46/iss1/10ON STRONG APPROXIMATION OF FUNCTIONS 159From this formula and (21) we get for h > 0 :kf ¡ fhkp · !(f ;h);(24)kf 0hkp · h¡1 !(f ;h);(25)i.e. fh 2 C1p if f 2 Cp. It is obvious thatjf(t)¡ f(x)j · jf(t)¡ fh(t)j+ jfh(t)¡ fh(x)j+ jfh(x)¡ f(x)jfor x; t 2 R0 and h > 0: This fact and (12), (8) and (9) and the Minkowskiinequality imply thatHqn(f ;A;x) · (Ln (jf(t)¡ fh(t)jq;A;x))1q ++ (Ln (jfh(t)¡ fh(x)jq;A;x))1q + jfh(x)¡ f(x)j:=3Xi=1Zn;i(x);for x 2 R0, n 2 N and h > 0. By (24) we havekZn;3( ¢ )kp · !(f ;h); h > 0:Applying (16) and (24), we getkZn;1( ¢ )kp ·M2(p; q; A) kf ¡ fhkp ·M2(p; q; A)!(f ;h); h > 0:By Theorem 3.1 and (25) we havewp(x)Zn;2(x) ·M5 kf 0hkp (Tn;2s(x;A))12s·M5 h¡1 !(f ;h) (Tn;2s(x;A))12s :From the above and by the property (iv) of A we obtainwp(x)Hqn(f ;A;x) ·M6(p; q; A)!(f ;h)µ1 + h¡11 + xpn¶:Setting h = 1+xpn, we obtain (23) for q ¸ 1:If 0 < q < 1 then by (14) we haveHqn(f ;A;x) · H1n(f ;A;x); x 2 R0; n 2 N;and by (23) for q = 1 we get (23) for 0 < q < 1: ¤Theorem 3.2 implies the followingCorollary 2. If the assumptions of Theorem 3.2 are satis¯ed, then for everyf 2 Cp, p 2 N0, we havelimn!1 Hqn(f ;A;x) = 0 at every x 2 R0:This convergence is uniform on every interval [x1; x2], x1 ¸ 0:7Rempulska and Skorupka: On Strong Approximation of Functions by Certain Linear OperatorsProduced by The Berkeley Electronic Press, 2004160 L. REMPULSKA AND M. SKORUPKARemark. The inequality (13) shows that results given for Hqn(f ;A) in Theo-rem 3.1, Theorem 3.2 and Corollary 2 concern also the di®erence jLn(f ;A;x)¡f(x)j. Thus the strong approximation for considered operators is moregeneral.3.2. Now we shall give three examples of operators of the Ln(f ;A) typede¯ned by (8).1. The Sz¶asz-Mirakyan operators Sn, n 2 N , de¯ned by (1) are generatedby the matrix A1 = [ank(x)] withank(x) = e¡nx(nx)kk!; n 2 N; k 2 N0; x 2 R0:It is easily veri¯ed that A1 2 ­, i.e. the A1 satis¯es the conditions (i) - (iv).2. The Baskakov operators with Vn, n 2 N , de¯ned by (2), are connectedwith the matrix A2 on the elementsank(x) =µn¡ 1 + kk¶xk (1 + x)¡n¡k; n 2 N; k 2 N0; x 2 R0:We can prove that A2 2 ­ also.3. The Bernstein operatorsBn(f ;x) :=nXk=0µnk¶xk (1¡ x)n¡k fµkn¶; n 2 N;de¯ned for continuous functions f on the interval [0; 1] are operators of thetype Ln(f ;A) with the matrix A3 = [ank(x)] whereank(x) =8><>:Ãnk!xk(1¡ x)n¡k if 0 · k · n;0 if k > n;for n 2 N . Here for considered functions f( ¢ ) and an( ¢ ) we set: f(x) = f(1)and ank(x) = ank(1) for all x > 1. We can verify that A3 2 ­.Hence the above lemmas, theorems and corollaries concern also the strongapproximation of functions by the Sz¶asz-Mirakyan, Baskakov and Bernsteinoperators.We remark also that the order of the strong di®erences given in Theorem3.2 and Corollary 2 are similar to (5) and (6) for the Baskakov operators.8Mathematical Journal of Okayama University, Vol. 46 [2004], Iss. 1, Art. 10http://escholarship.lib.okayama-u.ac.jp/mjou/vol46/iss1/10ON STRONG APPROXIMATION OF FUNCTIONS 161References[1] V. Baskakov, An example of sequence of linear operators in the space of continuousfunctions, Dokl. Akad. Nauk. SSSR, 113(1957), 249{251.[2] M. Becker, Global approximation theorems for Sz¶asz-Mirakjan and Baskakov opera-tors in polynomial weight spaces, Indiana Univ. Math. J., 27(1)(1978), 127{142.[3] R.A. De Vore, G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin1993.[4] Z. Ditzian, V. Totik, Moduli of Smoothness, Springer-Verlag, New York 1987.[5] L. Leindler, Strong Approximation by Fourier Series, Akad. Kiado, Budapest 1985.[6] O. Sz¶asz, Generalizations of S. Bernstein's polynomials to the in¯nite interval, J. Res.Nat. Bur. Standards, Sect. B., 45(1950), 239{245Lucyna RempulskaInstitute of Mathematics Pozna¶n University of TechnologyPiotrowo 3A 60-965 Pozna¶n, POLANDe-mail address: lrempuls@math.put.poznan.plMariola SkorupkaInstitute of Mathematics Pozna¶n University of TechnologyPiotrowo 3A 60-965 Pozna¶n, POLANDe-mail address: mariolas@math.put.poznan.pl(Received December 25, 2003 )9Rempulska and Skorupka: On Strong Approximation of Functions by Certain Linear OperatorsProduced by The Berkeley Electronic Press, 2004

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