Rate of convergence for Szasz–Mirakyan–Durrmeyer operators with derivatives of bounded variation
Introduction
Let DBγ(0, ∞),(γ ⩾ 0) be the class of all locally integrable functions defined on (0, ∞), with the growth condition ∣f(t)∣ ⩽ Meγt, M > 0 and f′ ∈ BV on every finite subinterval [0, ∞). Then for a function f ∈ DBγ(0, ∞).
We consider the Szasz–Mirakyan–Durrmeyer operators [4], which are defined bywhereAlternately we may rewrite (1) aswhereAlso let , then it is easily verified thatVery recently Srivastava et al. [5] and Gupta et al. [2] estimated the convergence rates for functions having derivatives of bounded variation for certain integral operators and second kind of beta operators respectively. Also very recently Gupta et al. [3] studied some other summation-integral type operators. In the present paper, we extend the results of Gupta et al. [2] and study the rate of convergence by means of the decomposition technique of functions having derivatives of bounded variation. More precisely the functions having derivatives of bounded variation on every finite subinterval on the interval [0, ∞) be defined aswhere ψ is a function of bounded variation on [a, b]. We denote the auxiliary function fx, by
Section snippets
Auxiliary results
In this section we give certain results, which are necessary to prove the main result. Lemma 1 [1]For m ∈ N ∪ {0}, if we define the mth order moment bythenand .
Also there holds the following recurrence relation:Consequently by the recurrence relation, for all x ∈ [0, ∞), we have Remark 1 If n ⩾ 2, then from Lemma 1, it is easily verified that Remark 2 It is also
Main result
In this section we prove the following main theorem. Theorem Let f ∈ DBγ(0, ∞), γ > 0 and x ∈ (0, ∞). Then for n ⩾ max{2, 4γ}, we havewhere denotes the total variation of fx on [a, b]. Proof We haveusing the identity
References (5)
- et al.
Rate of convergence in simultaneous approximation for Szasz–Mirakyan–Durrmeyer operators
J. Math. Anal. Appl.
(2006) - et al.
Convergence of a certain family of summation integral type operators
Appl. Math. Comput.
(2007)
Cited by (14)
On the integrated Baskakov type operators
2009, Applied Mathematics and ComputationApproximation on Durrmeyer modification of generalized Szász–Mirakjan operators
2024, Mathematical Methods in the Applied SciencesON APPROXIMATION PROPERTIES OF STANCU VARIANT λ-SZÁSZ-MIRAKJAN-DURRMEYER OPERATORS
2022, Korean Journal of Mathematics