Access the full text.
Sign up today, get DeepDyve free for 14 days.
N. Temme (2014)
Asymptotic Methods For Integrals
Song Li, Ren Wang (1999)
The characterization of the derivatives for linear combinations of Post-Widder operations in L pJournal of Approximation Theory, 97
B. Draganov, K. Ivanov (2007)
A characterization of weighted approximations by the Post-Widder and the Gamma operatorsJ. Approx. Theory, 146
V Miheşan (2008)
Gamma approximating operatorsCreative Math. Inf., 17
NISTDigital Library ofMathematical Functions
Melek Sofyalıoğlu, Kadir Kanat (2020)
Approximation properties of the Post‐Widder operators preserving e2ax,a>0Mathematical Methods in the Applied Sciences, 43
(1974)
Advanced Combinatorics
Vijay Gupta, P. Maheshwari (2019)
Approximation with certain Post-Widder operatorsPublications de l'Institut Math?matique (Belgrade)
Mourad Ismail, C.Ping May (1978)
On a family of approximation operatorsJournal of Mathematical Analysis and Applications, 63
Vijay Gupta, V. Singh (2019)
Modified Post-Widder Operators Preserving Exponential FunctionsAdvances in Mechanics and Mathematics
Philippe Laval (1991)
The laplace transformActa Applicandae Mathematica, 23
M. Siddiqui, R. Agrawal (2011)
A Voronovskaya Type Theorem on Modified Post-Widder Operators Preserving x 2Kyungpook Mathematical Journal, 51
M Sofyalıoğlu, K Kanat (2020)
Approximation properties of the Post–Widder operators preserving e2ax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{2ax}$$\end{document}, a>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0$$\end{document}Math. Methods Appl. Sci., 43
B. Draganov, K. Ivanov (2010)
A characterization of weighted approximations by the Post-Widder and the Gamma operators, IIJ. Approx. Theory, 162
Z. Ditzian, V. Totik (1987)
Moduli of smoothness
P. Sikkema (1970)
On the asymptotic approximation with operators of Meyer-König and Zeller, 73
P. Sikkema (1970)
On some linear positive operators, 73
(1973)
Linear combinations of linear positive operators and generating relations on special functions
L Comtet (1974)
10.1007/978-94-010-2196-8Advanced Combinatorics
Vijay Gupta, G. Tachev (2022)
Some Results on Post-Widder Operators Preserving Test Function x^rKragujevac Journal of Mathematics
In this paper, we study local approximation properties of certain gamma-type oper- ators. They generalize the Post–Widder operators and the Rathore operators, and approximate locally integrable functions satisfying a certain growth condition on the infinite interval [0, ∞). We derive the complete asymptotic expansion for these oper- ators and prove a localization result. Also, we estimate the rate of convergence for functions of bounded variation. Keywords Gamma type operators · Post–Widder operators · Generalized Rathore operators · Rate of convergence · Functions of bounded variation · Complete asymptotic expansion Mathematics Subject Classification 41A25 · 41A36 · 41A60 1 Introduction The Post–Widder operator plays a crucial role in the inversion of the Laplace transform. Let f : [0, ∞) → R be locally integrable and let L f denote its Laplace transform −sx (L f )(s) = e f (x ) dx . Communicated by Sergey Tikhonov. Ulrich Abel Ulrich.Abel@mnd.thm.de Vijay Gupta vijaygupta2001@hotmail.com Department MND, Technische Hochschule Mittelhessen, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany Department of Mathematics, Netaji Subhas University of Technology, Sector 3 Dwarka, New Delhi 110078, India 0123456789().: V,-vol 43 Page 2 of 15 U. Abel and V. Gupta If the integral converges for some s > 0, then the inversion formula f (x ) = lim L [L f ] n,x n→∞ is valid, for all positive x in the Lebesgue set of f [18, Chapter 7, Theorem 6a]. The operator L is defined by the equation n,x n+1 n n n (n) L [g] = (−1) g , n,x x x for any real positive number x and any positive integer n [18, Chapter 7, Definition 6]. A simple calculation reveals that n+1 1 n −nt /x n L [L f ] = e t f (t ) dt (x > 0) n,x n! x [18, Page 288]. For the sake of approximation, the Post–Widder operator P is defined in the slightly different form n ∞ (n/x ) −nt /x n−1 (P f )(x ) = e t f (t ) dt (x > 0) . (1.1) (n) [2, Eq. (9.1.9)] (cf. [10, Eq. (3.5)]). The form (1.1) is an operator of exponential type and these operators preserve linear functions [8]. The connection is as follows. Fix [x] −t /x x > 0. Define f (t ) = e f (t ). Then, n+1 n+1 ∞ n + 1 (n/x ) − n+1 t /x n ( ) (P f )(x ) = e t f (t ) dt n+1 n n! n+1 n + 1 [ ] = L L f . n,x Hence, in each Lebesgue point of f we have n+1 n + 1 [x] −x /x lim (P f )(x ) = lim L L f = e · e f (x ) = f (x ) . n+1 n,x n→∞ n→∞ The Post–Widder operators P were intensively studied by several authors [3, 4, 9]. In recent years, several authors defined and studied variants of the Post–Widder operator which preserve several test functions [5–7, 13, 16]. In order to include the similar operator by Rathore [12] (see below), we study in this paper a more general gamma type operator depending on a positive parameter, which includes both, the Post–Widder operators and the Rathore operators as special cases. Let E be the class of all locally integrable functions of exponential type on [0, +∞) At with the property | f (t )| ≤ Me (t ≥ 0) for some finite constants M , A > 0. The The rate of convergence of a generalization... Page 3 of 15 43 gamma-type operators P (cf. [10, Eq. (3.3) ]) associate to each f ∈ E the function n,c ncx ∞ (nc) −nct ncx −1 P f (x ) = e t f (t ) dt (x > 0) , (1.2) n,c (ncx ) where c is a positive parameter. We emphasize the fact that c may depend on the variable x. Note that the integral exists if nc > A. The definition can be rewritten in the form P f (x ) = φ (x , t ) f (t ) dt n,c n,c with the kernel function ncx (nc) −nct ncx −1 φ (x , t ) = e t . (1.3) n,c (ncx ) In the special case c = 1 these operators reduce to the Rathore operators R ≡ P , n n,1 given by [10, Eq. (3.6)] nx ∞ −nt nx −1 (R f )(x ) = e t f (t ) dt (x > 0) . (nx ) If we substitute c = 1/x, we obtain the Post–Widder operators (1.1). In this paper we derive the complete asymptotic expansion for the sequence of operators P in the form n,c −k P f (x ) ∼ f (x ) + a ( f , c, x ) n (n →∞), (1.4) n,c k k=1 provided that f admits derivatives of sufficiently high order at x > 0. Formula ( 1.4) means that, for all q = 0, 1, 2,..., there holds −k −q P f (x ) = a ( f , c, x ) n + o(n )(n →∞) n,c k k=0 where a ( f , c, x ) = f (x ). The coefficients a ( f , c, x ), which are independent of n, 0 k will be given in an explicit form. It turns out that associated Stirling numbers of the first kind play an important role. As a special case we obtain the complete asymptotic expansion for the Rathore operators R and for the Post–Widder operators P . n n Secondly, we study the rate of convergence of the sequence P f (x ) as n →∞ n,c for functions of bounded variation. More precisely, we present an estimate of the difference P f (x ) − ( f (x +) + f (x −)) /2. n,c 43 Page 4 of 15 U. Abel and V. Gupta 2 Main results For q ∈ N and x ∈ (0, ∞),let K [q; x] be the class of all functions f ∈ E which are q times differentiable at x. The following theorem presents as our main result the complete asymptotic expansion for the operators P . n,c Theorem 2.1 Let q ∈ N and x ∈ (0, ∞). For each function f ∈ K [2q; x],the operators P possess the asymptotic expansion n,c 2k k ( j ) −1 f x ( ) ( ) j −k −q P f x = f x + s j , j − k x + o n ( ) ( ) ( ) n,c 2 j ! (nc) k=1 j =k as n →∞, where s ( j , i ) denote the associated Stirling numbers of the first kind. The associated Stirling numbers of the first kind can be defined by their double generating function j −tu u s (i , j ) u = e (1 + t ) i ! i , j =0 (see [1, page 295, Ex. *20]). For q = 4, we obtain P f (x ) n,c (2) (3) 2 (4) xf (x ) 8xf (x ) + 3x f (x ) = f (x ) + + 2cn 24 (cn) (4) 2 (5) 3 (6) 12xf (x ) + 8x f (x ) + x f (x ) 48 (cn) (5) 2 (6) 3 (7) 4 (8) 1152xf (x ) + 1040x f (x ) + 240x f (x ) + 15x f (x ) 5760 (cn) −4 +o n as n →∞. In particular, we obtain the Voronovskaja-type formula (2) lim n P f (x ) − f (x ) = f (x ) , (2.1) n,c n→∞ 2c for f ∈ K [2; x]. In the special case c = 1 we have the complete asymptotic expansion for the Rathore operators, ∞ 2k ( j ) (−1) f (x ) j −k (R f )(x ) ∼ f (x ) + s ( j , j − k) x n 2 n j ! k=1 j =k The rate of convergence of a generalization... Page 5 of 15 43 as n →∞. In the special case c = 1/x we have the complete asymptotic expansion for the Post–Widder operators ∞ 2k k ( j ) (−1) f (x ) (P f )(x ) ∼ f (x ) + s ( j , j − k) x n 2 n j ! k=1 j =k as n →∞. Our second main result is an estimate of the rate of convergence for functions f ∈ E, which are of bounded variation (BV) on each finite subinterval of (0, ∞). Theorem 2.2 Let f ∈ E be a function of bounded variation on each finite subinterval of (0, ∞). Then, for each x > 0, we have the estimate f (x +) + f (x −) P f (x ) − n,c 1 1 cx + 2 x +x / k ≤ + O | f x + − f x − | + v f √ ( ) ( ) ( ) x −x / k n ncx 18πncx k=1 +O (exp (−βn)) as n →∞, where β = (1 − log 2) cx > 0 and the function f is defined as f (y) − f (x −) , 0 < y < x , f (y) = f (y) − f (x +) , x < y < ∞, 0, y = x . For the proofs of Theorems 2.1 and 2.2 we need a localization result for the operators P . Since it is interesting in itself we state it as a theorem. n,c Theorem 2.3 Let x ≥ δ> 0.If f ∈ E vanishes in a neighborhood (x − δ, x + δ) of x, then it exists a positive constant β such that P f (x ) = O (exp (−βcn)) (n →∞) . n,c The constant β can be chosen to be x + δ β = δ − x log > 0. Note that δ> x log ((x + δ) /x ) for x,δ > 0. 3 Auxiliary results and proofs Firstly, we study the moments of the operators P . Throughout the paper, let e n,c r denote the monomials, given by e (x ) = x (r = 0, 1, 2,...). Furthermore, define r 43 Page 6 of 15 U. Abel and V. Gupta ψ = e − xe ,for x ∈ R. In the following, the quantities denote the unsigned x 1 0 Stirling numbers of the first kind defined by m m− j j z = (−1) z (m = 0, 1, 2,...) , j =0 0 m where z = 1, z = z (z − 1) ··· (z − m + 1), m ∈ N, are the falling factorials. Using m m m (−z) = (−1) (z + m − 1) we obtain the relations m j (z + m − 1) = z (m = 0, 1, 2,...) . (3.1) j =0 We recall some known facts about Stirling numbers which will be useful in the sequel. The Stirling numbers of the first kind possess the representation 2m m r r m m = (−1) s (i , i − m) = (−1) s (i + m, i ) , 2 2 r − m i i + m i =m i =0 (3.2) for 0 ≤ m ≤ r,(see[1, page 226–227, Ex. 16]). The coefficients s (i , i − m), called associated Stirling numbers of the first kind, are independent of r. Lemma 3.1 The moments of the operators P are given by n,c r −k P e (x ) = x (r = 0, 1, 2,...) . n,c r r − k (nc) k=0 In particular, we have P e = e , P e = e and n,c 0 0 n,c 1 1 P e (x ) = x + , n,c 2 nc 3x 2x P e x = x + + , ( ) n,c 3 nc (nc) 3 2 6x 11x 6x P e (x ) = x + + + . n,c 4 2 3 nc (nc) (nc) Proof We have ncx ∞ (nc) (r + ncx ) −nct r +ncx −1 P e (x ) = e t dt = n,c r (ncx ) (ncx)(nc) (ncx + r − 1) = . (nc) The rate of convergence of a generalization... Page 7 of 15 43 Application of formula ( 3.1) yields P e (x ) = (ncx ) n,c r (nc) j =0 and the index transform j = r − k completes the proof. Lemma 3.2 The central moments of the operators P are given by n,c j j −k j −k x j r + k j −k−r P ψ (x ) = (−1) n,c x nc r + k ( ) k=0 r =0 ( j = 0, 1, 2,...). 0 1 In particular, we have P ψ (x ) = 1, P ψ (x ) = 0 and n,c n,c x x P ψ (x ) = x / (nc). n,c Proof Application of the binomial formula yields for the central moments j −r P ψ (x ) = (−x ) P e (x ) n,c x n,c r r =0 j j j −k x j j −r = (−1) r − k (nc) k=0 r =k and an index shift r → r + k yields the desired representation. Lemma 3.3 For each x > 0 and j = 0, 1, 2,..., the central moments of the operators − ( j +1)/2 P satisfy the relation P ψ (x ) = O n as n →∞. More precisely, n,c n,c x they have the representation j −k P ψ (x ) = (−1) s ( j , j − k) . n,c x 2 (nc) k= ( j +1)/2 Proof Taking advantage of the formula (3.2) we obtain j j −k j −k x j r + k j −r P ψ (x ) = (−1) s (i + k, i ) . n,c x 2 r + k i + k (nc) k=0 r =0 i =0 r +k j r +k j j −i −k Note that = 0, for i > r. Using the binomial identity = , i +k r +k i +k i +k r −i for 0 ≤ i ≤ r, we obtain j k j −k j −k x j j − k − i j −r P ψ (x ) = s (i + k, i ) (−1) . n,c x 2 i + k r − i (nc) k=0 i =0 r =i 43 Page 8 of 15 U. Abel and V. Gupta The inner sum is to be read as zero if i > j − k. Since j −k j −k−i ⎨ 0 (i < j − k) , j − k − i j − k − i j −r j −r −i (−1) = (−1) = r − i r r =i r =0 1 (i = j − k) , we conclude that j −k P ψ (x ) = (−1) s ( j , j − k) , n,c x 2 (nc) k= ( j +1)/2 which completes the proof. In order to derive Theorem 2.1, a general approximation theorem due to Sikkema ( ) [14, Theorem 3] (see also [15]) will be applied. For j ∈ N and x > 0, let H (x ) denote the class of all locally bounded real functions f : [0, ∞) → R, which are − j j times differentiable at x, and satisfy the additional condition f (t ) = O t as t →+∞. An inspection of the proof of Sikkema’s result reveals that it can be stated in the following form which is more appropriate for our purposes. Lemma 3.4 Let q ∈ N and let (L ) be a sequence of positive linear operators, n n∈N 2q+1 (2q) L : H (x ) → C [c, d],x ∈ [c, d]. Suppose that the operators L apply to ψ n n x 2q+2 and to ψ . Then the condition − ( j +1)/2 L ψ (x ) = O n (n →∞) , for j = 0, 1,..., 2q + 2, n x (2q) implies, for each function f ∈ H (x ), the asymptotic relation 2q ( j ) f (x ) −q (L f )(x ) = L ψ (x ) + o n (n →∞) . n n x j ! j =0 ( j ) In the application used in the proof of Theorem 2.1, we restrict H (x ) to consist only of locally integrable functions. We proceed with the proof of the localization result (Theorem 2.3), which will be applied in the proofs of Theorems 2.1 and 2.2. At Proof of Theorem 2.3 Let f ∈ E.From | f (t )| ≤ Me (t ≥ 0) we obtain the estimate ncx x −δ ∞ (nc) −(nc−A)t ncx −1 P f (x ) ≤ M + e t dt n,c (ncx ) 0 x +δ sx = M I + I , ( ) 1 2 (sx ) say, where s = nc > 0 and x −δ −(s−A)t sx −1 −sx I = e t dt = (s − A) γ (sx , (s − A)(x − δ)) , 0 The rate of convergence of a generalization... Page 9 of 15 43 −(s−A)t sx −1 −sx I = e t dt = s − A sx , s − A x + δ , ( ) ( ( )( )) x +δ where b ∞ −t z−1 −t z−1 γ (z, b) = e t dt, (z, b) = e t dt (Rez > 0, b ≥ 0) 0 b denote the lower and the upper incomplete gamma function, respectively. We use the well-known asymptotic behaviour of the incomplete gamma function for large parameters z and b. It holds z −b b e γ (z, b) ∼ , (3.3) (1 − λ) z [17, Eq. (7.3.18)], as z, b →∞ such that the ratio λ = b/z is bounded away from unity, i.e., λ ≤ λ < 1, where λ is a fixed number in (0, 1). In a similar kind it holds 0 0 z−1 −b b e (z, b) ∼ , (3.4) 1 − α [17, Eq. (7.4.43)], as z, b →∞ such that the ratio α = z/b is bounded away from unity, i.e., α ≤ α < 1. If δ = x the integral I vanishes. Let us consider the case δ< x. Since (s − A)(x − δ) x − δ λ = → < 1 (s →∞) sx x and 1 − λ = δ/x + A (x − δ) / (sx ),Eq. (3.3)impliesthat −1 sx −(s−A)(x −δ) I ∼ (δs + A (x − δ)) (x − δ) e (s →∞) . Application of Stirling’s formula, −z z−1/2 z ∼ 2πe z z →+∞ , ( ) ( ) leads to sx sx s M sx x − δ δs+A(x −δ) M I ∼ √ e (s →∞) . (sx ) δs + A (x − δ) x 2π Since x − δ e < 1 (0 <δ < x ) , x 43 Page 10 of 15 U. Abel and V. Gupta sx we conclude that M I = O (exp (−β s)) as s →∞, where β =−δ − 1 1 1 (sx ) x −δ x log > 0. Now we turn to the estimate of I . Since sx x α = → < 1 (s →∞) (s − A)(x + δ) x + δ δs−A(x +δ) and 1 − α = ,Eq. (3.4) implies that (s−A)(x +δ) sx −(s−A)(x +δ) I ∼ (x + δ) e (s →∞) . δs − A (x + δ) Application of Stirling’s formula leads to sx sx s M sx x + δ −δs+A(x +δ) M I ∼ √ · e (s →∞) . sx δs − A x + δ x ( ) ( ) 2π Since x + δ −δ e < 1 (x,δ > 0) , sx we conclude that M I = O (exp (−β s)) as s →∞, where β = δ − 2 2 2 (sx ) x x x +δ x −δ δ x +δ −δ x log > 0. Observe that β ≥ β because e ≤ e . 1 2 x x x 1+t The latter inequality is equivalent to the obvious inequality 2t ≤ log = 1−t 3 5 7 2 t + t /3 + t /5 + t /7 + ··· ,for t = δ/x ∈ [0, 1). Combining the above results we obtain the desired estimate with the constant β = β . Proof of Theorem 2.1 Let x > 0 and put U (x ) = (x − r , x + r ) ∩ [0, +∞),for (2q) r > 0. Let δ> 0 be given. Suppose that f (x ) exists. Choose a function ϕ ∈ C ([0, +∞)) with ϕ (x ) = 1on U (x ) and ϕ (x ) = 0on [0, +∞) \U (x ). δ 2δ ( j ) ( j ) Put f = ϕ f . Then we have f ≡ f on U (x ) which implies f (x ) = f (x ),for j = 0,..., 2q, and f ≡ 0on [0, +∞) \U (x ). By the localization theorem (Theo- 2δ rem 2.3), P f − f (x ) decays exponentially fast as n →∞. Consequently, f n,c and f possess the same asymptotic expansion of the form (1.4). Therefore, without loss of generality, we can assume that f ≡ 0on 0, +∞ \U x . By Lemma 3.3, [ ) ( ) 2δ 2 j − j we have P ψ (x ) = O n as n →∞. Under these conditions, Lemma 3.4 n,c x implies that 2q ( j ) f (x ) −q P f (x ) = f (x ) + P ψ (x ) + o n (n →∞) . n,c n,c x j ! j =1 The rate of convergence of a generalization... Page 11 of 15 43 By Lemma 3.3, we obtain 2q 2q j ( j ) ( j ) j −k f (x ) f (x ) x P ψ (x ) = (−1) s ( j , j − k) . n,c x 2 j ! j ! (nc) j =1 j =1 k= ( j +1)/2 Interchanging the order of summation, we obtain 2q min{2k,2q} k ( j ) (−1) f (x ) j −k −q P f (x ) = f (x ) + x s ( j , j − k) + o n n,c 2 j ! (nc) k=1 j =k as n →∞. Taking into account that 2q min{2k,2q} ( j ) (−1) f (x ) j −k −q x s ( j , j − k) = o n (n →∞) j ! (nc) k=q+1 j =k this implies the desired expansion (1.4) with the associated Stirling numbers of the first kind s (i , j ) as defined in Eq. (3.2). Now we turn to the estimate of the rate of convergence for BV functions. For the proof of Theorem 2.2 we apply the following properties of the kernel function φ (x , t ) as defined in (1.3). n,c Lemma 3.5 The kernel function φ (x , t ) satisfies the following estimates: n,c φ x , t dt ≤ 0 < y < x ( ) ( ) n,c nc (x − y) and φ (x , t ) dt ≤ (x < z < +∞) . n,c nc (z − x ) Proof Since x − t ≥ x − y > 0, for 0 ≤ t ≤ y < x,wehave y y x − t 1 φ (x , t ) dt ≤ φ (x , t ) dt ≤ P ψ (x ) n,c n,c n,c x − y (x − y) 0 0 = . nc (x − y) The second estimate ∞ ∞ t − x 1 φ (x , t ) dt ≤ φ (x , t ) dt ≤ P ψ (x ) n,c n,c n,c z − x (z − x ) z z nc (z − x ) 43 Page 12 of 15 U. Abel and V. Gupta is obtained in an analogous manner. Lemma 3.6 For fixed x > 0, 1 1 1 φ (x , t ) dt = + √ + O (n →∞) . n,c 2 n 3 2πncx Proof With s = ncx we have x ncx s 1 1 −u ncx −1 −u s−1 φ (x , t ) dt = e u du = e u du n,c (ncx ) (s) 0 0 0 (s) − (s, s) = . ( ) Following [11, Eq. 8.11.12], the (upper) incomplete gamma function satisfies the asymptotic relation πs 1 s−1 −s −1/2 (s, s) = s e − + O s (s →+∞) . 2 3 Together with Stirling’s formula 2π 1 s −s −2 (s) = s e 1 + + O s (s →+∞) s 12s we obtain πs 1 s−1 −s −1/2 s e − + O s (s) − (s, s) 2 3 ∼ 1 − √ s −s −1 (s) s e 2π/s 1 + O s π 1 −1 − + O s 3 s = 1 − √ −1 2π + O s 1 1 −1 − + O s 3 2πs = 1 − −1 1 + O s 1 1 1 1 = 1 − − √ + O 1 + O 2 s s 3 2πs 1 1 1 = + √ + O 2 s 3 2πs as s →+∞. Hence, for fixed x > 0, ncx 1 1 1 1 −u ncx −1 e u du = + √ + O (n →∞) . (ncx ) 2 n 3 2πncx The rate of convergence of a generalization... Page 13 of 15 43 Proof of Theorem 2.2 Let x ∈ (0, ∞). We start with the estimate P f (x ) − ( f (x +) + f (x −)) n,c | | ≤ f (x +) − f (x −) · P signψ (x ) + P f (x ) . n,c x n,c x Due to the fact that P preserve constant functions, we have n,c ∞ x x P signψ (x ) = − φ (x , t ) dt = 2 − φ (x , t ) dt . n,c x n,c n,c x 0 0 Thus, by Lemma 3.6,wehave 2 1 P signψ (x ) =− √ + O (n →∞) . n,c x 3 2πncx Next we estimate P f (x ) as follows: n,c x P f (x ) n,c x = φ (x , t ) f (t ) dt n,c x √ √ x −x / n x +x / n 2x ∞ = + + + φ (x , t ) f (t ) dt n,c x √ √ 0 x −x / n x +x / n 2x =: I + I + I + I . 1 2 3 4 Define η x , y = φ x , t dt . ( ) ( ) n,c n,c Integration by parts yields x −x / n I = f (t ) d η (x , t ) 1 x t n,c x −x / n x x = f x − √ η x , x − √ − η (x , t ) d ( f (t )) . x n,c n,c t x n n Since | f (t )| ≤ v ( f ),wehave x x x −x / n x x |I | ≤ v ( f ) · η x , x − √ + η (x , t ) d −v ( f ) . 1 x n,c n,c t x x −x / n 0 43 Page 14 of 15 U. Abel and V. Gupta Applying Lemma 3.5, and in the next step integrating by parts, we get x −x / n 1 x 1 x x | | I ≤ v ( f ) + d (−v ( f )) 1 x t x x −x / n 2 cx nc (x − t ) x −x / n x 1 1 x x = v ( f ) + 2 v ( f ) dt x x 2 3 nc x (x − t ) x 1 1 x x ≤ v ( f ) + v ( f ) x x 2 2 x −x / k nc x x k=1 ≤ v ( f ) . x −x / k ncx k=1 √ √ x +x / n Next for t ∈ x − x / n, x + x / n and by fact d η (x , t ) ≤ 1, we t n,c x −x / n conclude that x +x / k |I | ≤ v ( f ) . 2 x x −x / k k=1 Arguing analogously as in estimate of I ,wehave x +x / k |I | ≤ v ( f ) . 3 x ncx k=1 Since f ∈ E, the localization result (Theorem 2.3) applied with δ = x implies that I = O (exp (−βcn)) as n →∞ with the constant β = (1 − log 2) x > 0. Collecting the estimates of I , I , I , I , we get the desired result. 1 2 3 4 Acknowledgements The authors are very grateful to the anonymous referee for a thorough reading of the manuscript. The excellent advice led to a better exposition of the paper. Several valuable remarks helped to improve parts of the manuscript and to correct some typos. In particular, the referee provided the elegant estimate of the integral I in the proof of Theorem 2.2. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The rate of convergence of a generalization... Page 15 of 15 43 References 1. Comtet, L.: Advanced Combinatorics. Reidel Publishing Comp, Dordrecht (1974) 2. Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer Series in Computational Mathematics, vol. 9. Springer, New York (1987) 3. Draganov, B.R., Ivanov, K.G.: A characterization of weighted approximations by the Post–Widder and the Gamma operators. J. Approx. Theory 146, 3–27 (2007) 4. Draganov, B.R., Ivanov, K.G.: A characterization of weighted approximations by the Post–Widder and the Gamma operators. II. J. Approx. Theory 162, 1805–1851 (2010) 5. Gupta, V., Maheshwari, P.: Approximation with certain Post–Widder operators. Publ. Inst. Math. Nouv. Sér. 105(119), 131–136 (2019) 6. Gupta, V., Singh, V.K.: Modified Post–Widder operators preserving exponential functions. In: Singh, V.K., et al. (eds.) Advances in Mathematical Methods and High Performance Computing, Advances in Mechanics and Mathematics, vol. 41, pp. 181–192. Springer Nature, Cham (2019) 7. Gupta, V., Tachev, G.: Some results on Post–Widder operators preserving test function x . Kragujev. J. Math. 46, 149–165 (2022) 8. Ismail, M., May, C.P.: On a family of approximation operators. J. Math. Anal. Appl. 63, 446–462 (1978) 9. Li, S., Wang, R.T.: The characterization of the derivatives for linear combinations of Post–Widder operators in L . J. Approx. Theory 97, 240–253 (1999) 10. Mihesan, ¸ V.: Gamma approximating operators. Creative Math. Inf. 17(3), 466–472 (2008) 11. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V., Cohl, H.S., McClain, M.A. eds.: NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.1.6 of 06-30 (2022) 12. Rathore, R.K.S.: Linear combinations of linear positive operators and generating relations on special functions. Ph.D. Thesis, Delhi (1973) 13. Siddiqui, Md.A., Agrawal, R.R.: A Voronovskaya type theorem on modified Post–Widder operators preserving x . Kyungpook Math. J. 51, 87–91 (2011) 14. Sikkema, P.C.: On some linear positive operators. Indag. Math. 32, 327–337 (1970) 15. Sikkema, P.C.: On the asymptotic approximation with operators of Meyer–König and Zeller. Indag. Math. 32, 428–440 (1970) 2ax 16. Sofyalıoglu, ˘ M., Kanat, K.: Approximation properties of the Post–Widder operators preserving e , a > 0. Math. Methods Appl. Sci. 43, 4272–4285 (2020) 17. Temme, N.M.: Asymptotic Methods for Integrals. World Scientific, Hackensack (2015) 18. Widder, D.V.: The Laplace Transform. Princeton Mathematical Series. Princeton University Press, Princeton (1941)
Advances in Operator Theory – Springer Journals
Published: Jul 1, 2023
Keywords: Gamma type operators; Post–Widder operators; Generalized Rathore operators; Rate of convergence; Functions of bounded variation; Complete asymptotic expansion; 41A25; 41A36; 41A60
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.