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
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 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.
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 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.
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 + [] 3 + 1 where ωf C ) is the modulus of continuity of f defined in.1) and w ) = 1 1+. 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 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
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] [] + 5 + [3] 4 + [] 3 + 1 + 1 + [] 3 + 1. 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 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 = 16 1+ 1 ) 6[3] [] +)+ [] )+ 3 1 ) )
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 ) 16 1+ ) Ω f δ) From.) we can write ) { [n] P n ϕ 4 ) 4 [3] [] +[n] ) 6 [n] 3.) ) Ω f δ). 1+ 1 ) ) δ 4P n ϕ 4 ). + [] ) 4 [n] + 1 ) { 4 [3] [] +) 6 + [] ) 4 + 1 ) 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
11 POST-WIDDER OPERATORS BASED ON -INTEGERS 87 REFERENCES 1. Anastassiou G.A.; Aral A. Generalized Picard singular integrals Comput. Math. Appl. 57 9) 81 83.. Anastassiou G.A.; Gal S.G. Geometric and approimation properties of a comple Post-Widder operator in the unit disk Appl. Math. Lett. 19 6) 314 319. 3. Aral A. Pointwise approimation by the generalization of Picard and Gauss- Weierstrass singular integrals J. Concr. Appl. Math. 6 8) 37 339. 4. Aral A. On the generalized Picard and Gauss Weierstrass singular integrals J. Comput. Anal. Appl. 8 6) 46 61. 5. Aral A.; Gal S.G. -generalizations of the Picard and Gauss-Weierstrass singular integrals Taiwanese J. Math. 1 8) 51 515. 6. Aral A. On a generalized λ-gauss Weierstrass singular integrals Fasc. Math. 35 5) 3 33. 7. Aral A. On convergence of singular integrals with non-isotropic kernels Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 5 1) 83 93 ). 8. Atakishiyev N.M.; Atakishiyeva M.K. A -analog of the Euler gamma integral Theoret. Math. Phys. 19 1) 135 1334. 9. Ditzian Z.; Totik V. Moduli of Smoothness Springer Series in Computational Mathematics 9 Springer-Verlag New York 1987. 1. Draganov B.R.; Ivanov K.G. A characterization of weighted approimations by the Post-Widder and the Gamma operators J. Appro. Theory 146 7) 3 7. 11. Draganov B.R.; Ivanov K.G. A characterization of weighted approimations by the Post-Widder and the gamma operators II J. Appro. Theory 16 1) 185 1851. 1. Gadzhiev A.D. Theorems of the type of P.P. Korovkin type theorems Math. Zametki 1976) 781 786. 13. Gasper G.; Rahman M. Basic Hypergeometric Series Encyclopedia of Mathematics and its Applications 35 Cambridge University Press Cambridge 199. 14. Holhoş A. Uniform approimation in weighted spaces using some positive linear operators Stud. Univ. Babeş-Bolyai Math. 56 11) 413 4. 15. Ispir. On modified Baskakov operators on weighted spaces Turkish J. Math. 5 1) 355 365. 16. May C.P. Saturation and inverse theorems for combinations of a class of eponential-type operators Canad. J. Math. 8 1976) 14 15.
88 DIDEM AYDIN ALI ARAL and GÜLEN BAŞCANBAZ-TUNCA 1 17. Phillips G.M. Bernstein polynomials based on the -integers Ann. Numer. Math. 4 1997) 511 518. 18. Rathore R.K.S.; Singh O.P. On convergence of derivatives of Post-Widder operators Indian J. Pure Appl. Math. 11 198) 547 561. 19. Rempulska L.; Skorupka M. On strong approimation applied to Post-Widder operators Anal. Theory Appl. 6) 17 18.. Rempulska L.; Skorupka M. The Voronovskaya-type theorems for Post-Widder and Stancu operators Int. J. Pure Appl. Math. 38 7) 467 48. 1. Sikkema P.C. On some linear positive operators Nederl. Akad. Wetensch. Proc. Ser. A 73 - Indag. Math. 3 197) 37 337.. Ünal Z.; Özarslan M.A.; Duman O. Approimation properties of real and comple Post-Widder operators based on -integers Miskolc Math. Notes 13 1) 581 63. 3. Widder D.V. The Laplace Transform Princeton Mathematical Series Princeton University Press Princeton N.J. 1941. 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