Rate of convergence of exponential type operators related to $$p\left( x\right) =2x^{3/2}$$ p x = 2 x 3 / 2 for fu
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Rate of convergence of exponential type operators related to p (x) = 2x 3/2 for functions of bounded variation Ulrich Abel1
· Vijay Gupta2
Received: 25 March 2020 / Accepted: 5 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract The present article deals with the approximation of certain exponential type operators defined by Ismail and May. We estimate the rate of convergence of these operators for functions of bounded variation. Keywords Ismail–May operators · Rate of convergence · Bounded variation · Exponential type operators Mathematics Subject Classification 41A25 · 41A30
1 Introduction Ismail and May [9] studied exponential type operators L n . These operators have the form ∞ (1) kn (x, t) f (t) dt (L n f ) (x) = 0
with kernels kn (x, t) satisfying the partial differential equation n ∂ kn (x, t) = kn (x, t) (t − x) . ∂x p (x) The special case p (x) = 2x 3/2 leads to the kernel √ √ √ kn (x, t) = e−n x ne−nt/ x t −1/2 I1 2n t + δ (t) ,
B
(2)
Ulrich Abel [email protected] Vijay Gupta [email protected]; [email protected]
1
Technische Hochschule Mittelhessen Fachbereich MND, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany
2
Department of Mathematics, Netaji Subhas University of Technology, Sector 3 Dwarka, New Delhi 110078, India 0123456789().: V,-vol
123
188
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U. Abel, V. Gupta
where δ (·) denotes the Dirac delta function and I1 the modified Bessel function of first kind given by z 1+2k ∞ 2 I1 (z) = . k!Γ (k + 2) k=0
The corresponding operator [9, (3.16)] has the explicit representation ∞ √ √ √ e−nt/ x t −1/2 I1 2n t f (t) dt + f (0) , (Tn f ) (x) = e−n x n
(3)
0
where x ∈ (0, ∞). Note that the operators (3) can alternatively be written in the form
∞ √ nt n √ ∞ sk n x sk−1 √ f (t) dt + e−n x f (0) , (Tn f ) (x) = √ x x 0
(4)
k=1
where x ∈ (0, ∞) and sk (x) = e−x
xk . k!
We observe that the operators Tn are closely related to the well-known Phillips operators [10] (see [11, Eq. (1.1), case c=0]) given by (Pn f ) (x) = n
∞ k=0
∞
sk (nx)
sk−1 (nt) f (t) dt + e−nx f (0) .
0
√ If we substitute n in the Phillips operators by the real parameter n/ x, weimmediately get the operators (3) by Ismail and May in the form (4). More precisely, we have Pn/√x f (x) = (Tn f ) (x). However, unlike the operators Tn , the Phillips operators Pn are not exponential type operators. Many other integral type operators (see, e.g., [1,2,6,7]) including the Phillips operators are not of exponential type. Recently Gupta [5] gave some direct results including an estimate in terms of the second order modulus of continuity and a Voronovskaja-type result for these operators. The recent paper [8] contains a quantitative asymptotic formula in terms of the modulus of continuity with exponential growth, a Korovkin-type result for exponential functions and also a Voronovskaja-type asymptotic formula in the simultaneous approximation. The present paper deals with the rate of convergence of the operators Tn for functions of bounded variat
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