A large family of linear positive operators
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A large family of linear positive operators Vijay Gupta1 Received: 2 April 2019 / Accepted: 14 June 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In the present paper, we introduce a general family of linear positive operators, which contains a wide range of linear positive operators as special cases viz. Baskakov–Durrmeyer type operators, Phillips operators, Bernstein–Durrmeyer polynomials, Srivastava–Gupta operators, Baskakov–Szász type operators, Szász-Beta type operators, Lupa¸s-Beta operators and Lupa¸s–Szász type operators etc. We also find the difference estimate between such operators and the Mihe¸san operators. Keywords Linear positive operators · Convergence · Difference of operators · Modulus of continuity Mathematics Subject Classification 30E10 · 41A25 · 41A35
1 General operators In order to approximate integral functions, Srivastava–Gupta [24] proposed a general sequence of linear positive operators having same basis functions in summation and integration. Here we extend the studies and propose a further generalized operators, which include many well known operators of Durrmeyer type, hybrid operators and the Srivastava–Gupta operators [24] (see also [18]), as special cases. For x ∈ [0, ∞), we introduce the following general family of linear positive operators Vn,α,β ( f , x) =
∞
β
m αn,k (x)G n,k ( f ),
(1)
k=0
where β G n,k ( f )
B 1
=
⎧ ⎨n ⎩
∞ 0
f (0),
β+1
m n,k−1 (t) f (t)dt,
1≤k 0, φn (x) = e−nx for c = 0. It is seen that if φn (x) = (1 + cx)−n/c , then one may get the operators (2) as ∞ n (cx)k k c k . f L n,c ( f , x) = n +k k (1 + cx) c n k=0
After a gap of twenty eight years in 2008 Mihe¸san [21] applied gamma transform to Szász operators and obtained the following sequence of linear positive operators Mn,γ ( f , x) =
∞
γ
m n,k (x)Fn,k ( f ),
(3)
k=0
where γ m n,k (x)
(γ )k . = k!
1+
nx γ
k
nx γ
γ +k
(γ )k · γ γ (nx)k = , Fn,k ( f ) = f k! (γ + nx)γ +k
k . n
The two approaches to construct the general linear positive operators are different, but the operators obtained are same as we see that two forms are connected by the relation γ = n/c. Also, in the approach of Mihe¸san, if γ = nx one may obtain the Lupa¸s operators (see [2]), which as such is not exponential type operator. All cases indicated in both of the unified approaches reproduce the linear functions. The motivation of present paper is based on these two papers by Mastroianni and Mihe¸san.
123
V. Gupta
2 Moment estimation Moment estimate is one of the important features concerning convergence of linear positive operators. Very recently in [13], we presented moments of many operators of discrete and integral type. For the operators (1) we give the following unified approach to determine moments, using hypergeometric function. The moments of different special cases of these operators can be obtained by assigning the values to α, β as indicated above. Lemma 1 The r -th order moment Vn,α,β (er , x), er = t r , r = 0, 1, 2, . . . satisfy the following repres
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