Jain-Baskakov Operators and its Different Generalization
- PDF / 607,582 Bytes
- 19 Pages / 439.642 x 666.49 pts Page_size
- 91 Downloads / 186 Views
Jain-Baskakov Operators and its Different Generalization Prashantkumar Patel · Vishnu Narayan Mishra
Received: 27 July 2013 / Revised: 25 October 2013 / Accepted: 9 December 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014
Abstract The aim of this work is to introduce a new sequence of positive linear operators, which is an integral version of Jain operators. We studied some direct results in ordinary approximation. Weighted approximation, rate of convergence, and A-statistical convergence are investigated. Also, we discussed approximation properties of King-type and Stancutype generalization of these new sequences of linear positive operators and investigated a modification of Jain-Baskakov operators with parameter c. At the end, we propose a qanalogue of Jain operators based on a q-integer. Keywords Jain operators · Baskakov operators · King-type operators · Stancu operators · Rate of convergence · Weighted approximation · A-Statistical convergence Mathematics Subject Classification (2010) Primary 41A25 · 41A30 · 41A36
1 Introduction For λ ≥ 0, consider the weight function ρλ : R+ → [1, ∞], ρλ (x) = 1 + x 2+λ . Define the space f (x) is convergent as x → ∞ . Cρλ (R+ ) = f ∈ C(R+ ) : ρλ (x)
P. Patel () · V. N. Mishra Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat 395 007, India e-mail: [email protected] P. Patel Department of Mathematics, St. Xavier’s College (Autonomous), Ahmedabad, Gujarat 380 009, India V. N. Mishra L. 1627 Awadh Puri Colony Beniganj, Phase -III, Opposite - Industrial Training Institute (I.T.I.), Ayodhya Main Road, Faizabad 224 001, Uttar Pradesh, India
P. Patel, V. N. Mishra
The norm on Cρλ (R+ ) is defined by |f (x)| . x≥0 ρλ (x)
f ρλ = sup
For β ∈ [0, 1), x ∈ R+ , and f ∈ Cρ0 (R+ ), we introduce a new sequence of positive linear operators, calling it Jain-Baskakov operators as follows: Knβ (f, x)
= (n − 1) =
∞
ωβ (v, nx)
∞
pn,v−1 (t)f (t)dt + e−nx f (0)
0
v=1 ∞
(1.1)
Wn (x, t)f (t)dt, 0
where
n+v−1 tv , pn,v (t) = v (1 + t)n+v
ωβ (v, nx) = nx(nx + vβ)v−1
e−(nx+vβ) v!
and Wn (x, t) = (n − 1)
∞
ωβ (v, nx)pn,v−1 (t) + e−nx δ(t),
v=1
with δ(t) being the Dirac delta function. The operators defined by (1.1) are the integral modification of Jain operators having weight functions of Baskakov basis function. As a special case, i.e., β = 0, the operators (1.1) reduce to the well-known Szasz-Baskakov operators. In 1972, Jain [18] introduced and studied the following class of positive linear operators: Pnβ (f, x) =
∞
ωβ (v, nx)f
v=0
v n
,
(1.2)
where β ∈ [0, 1) and f ∈ C(R+ ). In the particular case β = 0, Pn0 , n ∈ N, turns into well-known Szasz-Mirakjan operators [28]. We mention that a Kantorovich-type extension β β of Pn was given in [30]. Very recently, integral modification of the operators Pn having a weight function of some Beta basis functions was discussed in [24, 29]. The present pap
Data Loading...
