A GENERALIZATION OF POST-WIDDER OPERATORS

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI S.N.) MATEMATICĂ Tomul LXII 16 f.1 A GENERALIZATION OF POST-WIDDER OPERATORS BASED ON -INTEGERS BY DIDEM AYDIN ALI ARAL and GÜLEN BAŞCANBAZ-TUNCA Abstract. In this paper we introduce a -generalization of Post-Widder operators P n. We give a Voronovskaja-type approimation result and rate of that convergence. We also study approimation properties of P n in a weighted space. We also show that the rates of convergence of these generalized operators to approimating function f as weight are at least so faster than that of the classical Post-Widder operators. Mathematics Subject Classification 1: 4B 41A35. Key words: -Post-Widder operator Voronovskaja s theorem weighted modulus of continuity. 1. Introduction The Post-Widder operator is defined for all n N > by P n f)) = 1 n! n ) n+1 t n e nt f t)dt for f C ). The convergence of Post-Widder operators was investigated deeply in Ch. VII of [3]) by Widder. The saturation and inverse problems for Post-Widder operators of the form P n f)) = 1 n 1)! n ) n t n 1 e nt f t)dt were presented by May in[16]. Simultaneous approimation with an asymptotic formula of these operators were carried out by Rathore and Singh[18]. A modification of Post-Widder operators in polynomial weighted spaces of

2 78 DIDEM AYDIN ALI ARAL and GÜLEN BAŞCANBAZ-TUNCA differentiable functions and strong approimation was studied by Rempulska and Skorupka in [19]. On the other hand Voronovskaja-type results were discussed by the same authors in []. A deep analysis related to the characterization of the weighted approimation errors of the Post-Widder and the Gamma operators for functions belonging L p )1 p with a suitable weight was achieved by Draganov and Ivanov in [1]. Uniform approimation in a weighted space by the Post-Widder operators was obtained by Holhoş in [14]. Approimation and geometric properties of comple Post-Widder operators were studied in []. As it is known in 1997 an interesting contribution to the theory of approimation by linear positive operators was done by Phillips by epressing Bernstein polynomials in terms of -calculus. [17]. After linear positive discrete and integral operators were designed by using machinery of -calculus and investigated their approimation properties by many authors. Some are in [1] [3] [4] [5] [6] [7] [17]. In this work we introduce a -generalization of the Post-Widder operator denoted by P n n N and give a Voronovskaja-type result for P n. Net we study the rate of this convergence. On the other hand we deal with the approimation by P n in polynomial weighted space using the theorem proved by Gadziev in [1]. For the rate of the approimation we use the weighted modulus of continuity introduced by İspir in [15]. Throughout the paper related to -special functions we shall use the following standard notations of -numbers see [13]): Let > then -integer is defined as { 1 n 1 [n] = R+ \{1 n = 1 for n N = {1... and [] =. For < < 1 the -etension of eponential function e is defined by 1.1) E ) := n= nn 1) ; ) n n = ; ) R where a; ) n = n 1 k= 1 ak ) and ; ) = k= 1+k ). Also -factorial is defined by { [n] [n]! = [n 1] [1] n N 1 n =.

3 3 POST-WIDDER OPERATORS BASED ON -INTEGERS 79 In [8] Atakishiyev and Atakishiyeva introduced the -etension of the Euler Gamma integral as 1.) c )Γ ) = 1 1 1) ln t 1 dt Re > E 1 )t) for < < 1 where Γ ) is the -etension of Gamma function defined by 1.3) Γ ) = ; ) ; ) 1 ) 1 < < 1 and satisfies the analogous property Γ n+1) = [n]! for n N {. Moreover the factor c ) in 1.) satisfies the following conditions: a. c +1) = c ) b. c n) = 1 n = 1... c. lim 1 c ) = 1. Below taking 1.) into consideration we introduce a -etension of the Post-Widder operator. Definition 1. Let < < 1 n N and > the -Post-Widder operator will be defined by 1.4) P n f)) = A n Γ n) ) [n] n E t n 1 1 ) [n]t )f n t)dt for all real valued measurable functions defined on ) such that the integral in 1.4) is convergent for almost all where A n := and Γ n) is the -gamma function given in 1.3). nn 1) 1 ) ln 1 Lete α ) := α α [ ) >.NotethatthePost-Widderoperator P n satisfies P n e α )) = Γn+α) n α Γn) α whereγis thefamiliar gamma function. Γn+α) Moreover we have lim n n α Γn) = 1 see for eample in the proof of Lemma.1 of [14]). Analogously in the following lemma we give the behavior of -Post- Widder operator on the power function e α α.

4 8 DIDEM AYDIN ALI ARAL and GÜLEN BAŞCANBAZ-TUNCA 4 Lemma 1. For all e α ) = α α [ ) > we have P n e )) = 1 and 1.5) P n e α )) = c n+α) α α 1)Γ n+α) [n] α Γ n) α where Γ is the -gamma function given by 1.3). Proof.By the definition 1.) it is clear that P n e )) = 1. For α [ ) we have 1 )nn 1) P n e α )) = ln 1 Γ n) ) [n] n nα t n+α 1 E 1 ) [n] t )dt. Making use of substitution [n] t = u gives that P n e α )) = 1 )nn 1) +nα ln 1 Γ n) [n] ) α u n+α 1 E 1 )u) du from which we can reach to the result by taking 1.) into consideration.. Voronovskaja-type results In this section we first obtain a Voronovskaja-type result for P n. Subseuently westudytherateofthisconvergence inasimilarway tothatforthe classical Post-Widder operator obtained by Rempulska and Scorupka in []. So by considering the weight function w r ) := 1 1+ for r 1 and r w ) := 1 we similarly take C r r N as the space of all real valued functions on ) for which w r f is continuous and bounded on ) with norm f r = sup > w r ) f ). Moreover we simply take C as the space of all functions continuous and bounded on ). We also consider Cr r N := N { as the set of all functions f C r which are twice differentiable on ) and f k) C r k for k = 1 and C r k C if r k. The modulus of continuity of a function f C r is defined as.1) ωf;c r ;δ) := sup f +h) f ) r. h δ Let ϕ := t. In the following we give P n ϕ m )) for m = 14.

5 5 POST-WIDDER OPERATORS BASED ON -INTEGERS 81 Lemma. Let P n be the operator defined by 1.4). Then we have: P n ϕ )) = P n ϕ ) ) = [n] P n ϕ 4 ) ) = { [3] [] 6 [n] 3 + [3] 5 [n] + [] 4 [n] + 1 ) 3 4. [n] Proof.From 1.5) P n ϕ )) = is clear. For each k = 13 taking the fact [n+k] k [n] = [k] and the property b. of c into consideration we obtain that ) ) ) P n ϕ Γ n+) [n+1] ) = [n] Γ n) 1 = 1 = [n] [n] and.) ) P n ϕ 4 ) { Γ n+4) = 6 [n] 4 Γ n) 4 Γ n+3) 3 [n] 3 Γ n) +6 Γ n+) [n] Γ n) 3 = 1 6 [n] 3 {[3] [n+] [n+1] 3 3 [n] [] +[n] ) 4 = 1 {[3] [] +[n] )+ [n] [] )+ 3 [n] 1 ) 4 = 6 [n] 3 { [3] [] 6 [n] 3 + [3] 5 [n] + [] 4 [n] + 1 ) 3 4 [n] 4 which completes the proof. Remark 1. We point out that our -generalization P n f) n N is different by -analogue of the Post-Widder operators recently introduced in [] as follows. ) 1/1 ) Pn f)) = t f t n E t)d t [n] where [ ) and 1). Since P n ϕ )) = P n ϕ )) = [n] and Pnϕ )) = [n+1] [n] 1) Pnϕ )) = [n+1][n+] [n] [n+1] [n] +1) the behavior of P n f) on ϕ and ϕ are better than Pnf) on ϕ and ϕ.

6 8 DIDEM AYDIN ALI ARAL and GÜLEN BAŞCANBAZ-TUNCA 6 Using Lemma we reach to the following result easily. Lemma 3. Let n ) be a seuence in 1) such that lim n n = 1. Then for every > we have lim [n] n n P nn ϕ )) = lim [n] ) n n P nn ϕ ) = lim [n] ) n n P nn ϕ 4 ) =. Below we present a Voronovskaja-type result for the -Post-Widder operator P n. Theorem 1. Let n ) be a seuence in 1) such that lim n n = 1 and f C. Then we have lim n [n] n {P nn f)) f) = f ) at every >. Proof.Proof follows from Lemma 3 and Theorem 1 of [1]. Now we study the rate of this convergence by etending the result of Rempulska and Scorupka for the Post-Widder operator Theorem of section 3. of []) to the -Post-Widder operator P n. Theorem. Let f Cr with a fied r = 1. Then w ) [n] {P nf)) f ) f ) ) 1 {[3] f ω [] +) ;C ; [n] 5 + [] where ωf C ) is the modulus of continuity of f defined in.1) and w ) = Proof.Let f Cr and > be fied. From Taylor s formula with integral form of the remainder we have.3) f t) = f )+f )t )+ 1 f )t ) +t ) F t) for t > where.4) F t) = 1 1 u)[f +ut )) f )]du.

7 7 POST-WIDDER OPERATORS BASED ON -INTEGERS 83 Taking ϕ = t and using Lemma we obtain from.3) that w ) [n] {P nf)) f ) ).5) w )[n] P n ϕ F.) ) ) f where we have used the fact that P n is linear and positive. Recall the followingpropertyofthemodulusofcontinuityωf;c ;t) 1+ t )ωf;c δ ;δ) for f C. Using this we obtain the following estimate for F t) : 1 ) F t) 1 u)ω f ;C ;u t du ) 1 ω f ;C ; t 1 u)du 1 ω f ;C ; 1+ [n] ) [n] t ). Taking the last ineuality into account we deduce.6) [n] P n ϕ F.) ) ) f 1 ω ;C ; ){[n] P n ϕ ))+ [n] [n] P nϕ 4 )). Here from Lemma we find that.7) [n] P n ϕ ) ) = on the other hand since [n] < 1 1 we obtain that [n] P ) n ϕ 4 ) = { ) ) 5 [3] [n] [] +[n] + [n] [] + 3 [n] 1 ) { [3] = [] 5 + [3] [n] 4 + [] 3 + [n] 1 ) < { [3] [] 5 + [3] 4 + [] 3 + 1

8 84 DIDEM AYDIN ALI ARAL and GÜLEN BAŞCANBAZ-TUNCA 8 which together with.6) and.7) we reach to w )[n] P n ϕ F.) ) ) < 1 ω f ;C ; [n] = 1 ω f ;C ; [n] { [3] [] 5 ) [3] [] [3] 4 + [] [] Using the last ineuality in.5) the result follows. Recall that a continuous function f from D R into R is said to be Lipschitz continuous of order α α 1] if there eists a constant M > such that for every y Df satisfies f ) f y) M y α. The set of Lipschitz continuous functions is denoted by Lip α M D). Remark. Let f Cr for r = 1 such that f Lip α M ). Then we have w ) [n] {P nf)) f ) f ) +α 4 < M α { α+) Bα) [3] [] 4 + [3] where B..) is the usual beta function. 3 + [] + 1 ) [n] ) α 3. Approimation properties in a weighted space In this section by using a Bohman-Korovkin type theorem proved in [1] by Gadzhiev we present a direct approimation property of the operator P n. Let B denote the space of real-valued functions f defined on ) with w ) f ) M f for all > where M f is a constant depending on thefunctionf.thereforec can beregardedas asubspaceofb namely C = {f B : w f is continuous on ). We also consider the space of functions C k = {f C : lim w )f ) = k R > with the norm f = sup > w ) f ). Related to the spaces given above we recall the following result due to Gadziev in [1].

9 9 POST-WIDDER OPERATORS BASED ON -INTEGERS 85 Theorem 3. Let T n be a seuence of linear positive operators mapping C into B and satisfying the conditions lim n T n e i ) e i = for i = 1. Then for any f C k we have lim n T n f) f = and there eists a function f C \C k such that lim n T n f ) f 1. Now for f C k we will consider the weighted modulus of continuity f+h) f) defined by Ω f δ) = sup > h δ in [15]. This function has 1+ )1+h ) the following properties: 1. Ω f δ) f. Ω f mδ) m 1+δ ) Ω f δ) m N 3. lim δ Ω f δ) =. Note that we can not find a rate of convergence in terms of usual first modulus of continuity ωf;c ;δ) of function f because the modulus of continuity ωf;c ;δ) on the infiniteinterval does not tend to zero as δ. For this reason we consider the weighted modulus of continuity Ω fδ). Remark 3. Since any linear and positive operator is monotone the relations 1.5) guarantee that P n f) C for each f C. Theorem 4. Let n ) be a seuence in 1) such that lim n n = 1. Then for each f C k we have lim n P nn f) f =. Proof.From Lemma it is clear that lim n P nn e i ) e i = for i = 1. For i = we get P nn e )) e ) P nn e ) e = sup > 1+ sup > 1+ ) [n+1]n [n] n n [n] n = 1 n [n] n which implies that lim n P nn e ) e =. Since the conditions of Theorem3aresatisfied thenweobtainforanyf C k lim n P nn f) f =. Theorem 5. For f C k n N we have P n f)) f ) ) sup > 1+ ) 3 K Ω f [n] 1/4 where K = ) 6[3] [] +)+ [] )+ 3 1 ) )

10 86 DIDEM AYDIN ALI ARAL and GÜLEN BAŞCANBAZ-TUNCA 1 is independent of f and n. Proof.From the properties of Ω it is obvious that for any λ > Ω fλδ) λ+1) 1+δ ) Ω f δ). For < δ 1 if we use the definition of Ω and the last ineuality with λ = t δ we have f t) f ) 1+ )1+t ) )Ω f t ) 1+ )1+t ) ) 1+δ ) 1+ t δ Applying P n to the last ineuality we obtain 3.1) P n f)) f ) ) Ω f δ) From.) we can write ) { [n] P n ϕ 4 ) 4 [3] [] +[n] ) 6 [n] 3.) ) Ω f δ) ) ) δ 4P n ϕ 4 ). + [] ) 4 [n] + 1 ) { 4 [3] [] +) 6 + [] ) ) 3 for all > and n N. Taking 3.1) and 3.) into account and by choosing δ = [n] 1/4 we obtain that P nn f)) f ) ) sup > 1+ ) 3 K Ω f [n] 1/4 where K is the given in the hypothesis which is independent of f and n. Remark 4. Let be fied. Since [n] 1 1 as n we cannot P obtain that lim n sup nn f)) f) > = in Theorem 5. But we can 1+ ) 3 improve this by choosing = n where < n < 1 with n 1 as n. Actually in this case we obtain that lim n [n] n =. Therefore we have lim n Ω f [n] 1/4 n ) = and lim n K n is finite which show that we obtain a weighted approimation for P nn f)) to f) as n in Theorem 5. This theorem tells us that depending on the selection of n the rate of convergence of P nn f)) to f ) as weight is [n] 1/4 n that is at least so faster than n 1/4 which is the rate of convergence for the classical Post-Widder operators. On the other hand the saturation rate of the Post- Widder operator in L p norm 1 p with power-type weight is 1 n see chapter 1 of [9] or [11]). 3

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12 88 DIDEM AYDIN ALI ARAL and GÜLEN BAŞCANBAZ-TUNCA Phillips G.M. Bernstein polynomials based on the -integers Ann. Numer. Math ) Rathore R.K.S.; Singh O.P. On convergence of derivatives of Post-Widder operators Indian J. Pure Appl. Math ) Rempulska L.; Skorupka M. On strong approimation applied to Post-Widder operators Anal. Theory Appl. 6) Rempulska L.; Skorupka M. The Voronovskaya-type theorems for Post-Widder and Stancu operators Int. J. Pure Appl. Math. 38 7) Sikkema P.C. On some linear positive operators Nederl. Akad. Wetensch. Proc. Ser. A 73 - Indag. Math ) Ünal Z.; Özarslan M.A.; Duman O. Approimation properties of real and comple Post-Widder operators based on -integers Miskolc Math. Notes 13 1) Widder D.V. The Laplace Transform Princeton Mathematical Series Princeton University Press Princeton N.J Received: 15.VI.1 Revised: 7.XII.1 Accepted: 7.I.13 Ankara University Faculty of Science Department of Mathematics 61 Tandogan Ankara TURKEY didemaydn@hotmail.com Kırıkkale University Faculty of Science and Arts Department of Mathematics Yahşihan Kirikkale TURKEY aliaral73@yahoo.com Ankara University Faculty of Science Department of Mathematics 61 Tandogan Ankara TURKEY tunca@science.ankara.edu.tr