On approximation of certain integral operators
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ON APPROXIMATION OF CERTAIN INTEGRAL OPERATORS Vijay Gupta · Rani Yadav
Received: 4 December 2012 / Accepted: 13 February 2013 / Published online: 13 March 2014 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014
Abstract In the present paper we introduce the summation-integral type modified Lupas operators with weights of Beta basis functions. We define the operators in terms of hypergeometric series and, using such an approach, establish moments. Our main results are an asymptotic formula and an error estimate in terms of modulus of continuity and weighted approximation and rate of convergence for functions having bounded derivatives. Keywords Lupas operators · Rate of convergence · Asymptotic formula · Weighted approximation Mathematics Subject Classification (2000) 41A25 · 41A35
1 Introduction In 1995, Lupas [9] introduced the operator Ln (f, x) for f : [0, ∞) → R and x ≥ 0 as Ln (f, x) =
∞ k=0
where ln,k (x) = 2
−nx
ln,k (x)f
k , n
(1)
nx + k − 1 −k 2 . k
In order to approximate the integrable functions, the most common integral modifications of the discrete operators are the Kantorovich- and Durrmeyer-type modifications. Agra-
B
V. Gupta ( ) · R. Yadav Department of Mathematics, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 110078, India e-mail: [email protected] R. Yadav e-mail: [email protected]
194
V. GUPTA, R. YADAV
tini [1] obtained some approximation properties and an asymptotic formula for the Lupas– Kantorovich operators. The Durrmeyer variant of the operators (1) introduced in [1] is defined by ∞ ∞ Mn (f, x) = cn,k ln,k (x) ln,k (u)f (u) du, (2) 0
k=0
where cn,k = ∞ 0
1 . ln,k (t) dt
For er (t) = t , operators (2) satisfy the following relation: r
k (i+r)! i ∞ ln,k (x) i=0 (−1) sk,i (log 2)i , Mn (er , x) = nr (log r)r ki=0 (−1)i sk,i (i)! i k=0 (log 2)
where sk,i are the Stirling numbers of first kind, which are related to the Pochhammer symbol (x)k as (x)k = x(x + 1)(x + 2) · · · (x + k − 1) =
k (−1)k−i sk,i x i . i=0
In the recent years, some Durrmeyer-type operators were discussed in [2–4, 7, 8]. For f ∈ L1 [0, ∞), we now propose a new integral modification of the Lupas operators having weights of Beta basis function as Dn (f, x) =
∞
∞
ln,k (x)
bn,k−1 (t)f (t) dt + ln,0 (x)f (0),
x ∈ [0, ∞),
0
k=1
where ln,k (x) is given by (1), and bn,k (t) =
tk 1 B(n, k + 1) (1 + t)n+k+1
are the Beta basis functions. The hypergeometric function is defined as 2 F1 (a, b; c; x) =
∞ (a)k (b)k k=0
(c)k k!
xk .
The operators (3) can be written in terms of hypergeometric form: Dn (f, x) =
∞
ln,k (x)
bn,k−1 (t)f (t) dt + ln,0 (x)f (0)
0
k=1
= 2−nx
∞
∞ nx + k − 1 k=1
k
2−k
0
∞
t k−1 1 f (t) dt + 2−nx f (0) B(n, k) (1 + t)n+k
(3)
ON APPROXIMATION OF CERTAIN INTEGRAL OPERATORS
= 2−nx
∞
∞ t k−1 f (t) (nx + k − 1)!2−k (n + k − 1)! dt + 2−nx f (0) (1 + t)n k=1 k!(nx − 1)! (n − 1)!(k − 1)! (1 + t)k
∞
∞ f (t) (nx
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