Abstract
In this article, we introduce a new Durrmeyer-type generalization of p , q -Szász-Mirakjan operators using the p , q -gamma function of the second kind. The moments and central moments are obtained. Then, the Voronovskaja-type asymptotic formula is investigated and point-wise estimates of these operators are studied. Also, some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre K -functional. Finally, the rate of convergence and weighted approximation of these operators are presented.
Highlights
In recent years, ðp, qÞ-analogues of well-known positive operators have been widely constructed and researched sinceMursaleen et al first introduced ðp, qÞ-Berstein operators [1]and ðp, qÞ-Bernstein-Stancu operators [2]
In [8], Mursaleen et al introduced a new modification of Szász-Mirakjan operators based on the ðp, qÞ
Quantitative estimates for the convergence in the polynomial weighted spaces and Voronovskaya theorem for new ðp, qÞ-Szász-Mirakjan operators (1) were given
Summary
Ðp, qÞ-analogues of well-known positive operators have been widely constructed and researched since. In [7], Aral and Gupta constructed ðp, qÞ-Szász-Mirakjan-Durrmeyer operators using ðp, qÞ-gamma function of the first kind and estimated moments and established some direct results of these operators. Kantorovich-type ðp, qÞ-Szász-Mirakjan operators and discussed their error estimated. Quantitative estimates for the convergence in the polynomial weighted spaces and Voronovskaya theorem for new ðp, qÞ-Szász-Mirakjan operators (1) were given. All these achievement motivates us to construct the Durrmeyer analogue of the ðp, qÞ-Szász-Mirakjan operators defined by (1). Dm u ; t = 1⁄2mp,q 〠 sm,k ðt Þ q−k pk−m sm,k ðuÞu2 dp,q u ep,q −1⁄2mp,q pk−m+1 q−k u dp,q u.
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