Abstract
In this paper, we introduce a new type of (p; q) exponential function with some properties and a modified (p; q)-Szasz-Mirakyan operators by virtue of this function by investigating approximation properties. We obtain moments of generalized (p; q)-Szasz-Mirakyan operators. Furthermore, we derive direct results, rate of convergence, weighted approximation result, statistical convergence and Voronovskaya type result of these operators with numerical examples. Graphical representations reveal that modified (p; q)-Szasz-Mirakyan operators have a better approximation to continuous functions than pioneer one.
Highlights
Approximation theory is one of the oldest branches of mathematics
We introduce a new type of (p; q) exponential function with some properties and a modified (p; q)-Szasz-Mirakyan operators by virtue of this function by investigating approximation properties
Graphical representations reveal that modified (p; q)-Szasz-Mirakyan operators have a better approximation to continuous functions than pioneer one
Summary
Approximation theory is one of the oldest branches of mathematics. To approximate continuous functions with q-analogue of linear positive operators is significant application of q-calculus in approximation theory. (p; q)-Szasz-Mirakyan operators; uniform convergence; Voronovskaya type result; statistical convergence. Uniform convergence and Voronovskaya result of the above operators. We define a different sort of modified (p, q)-Szasz-Mirakyan operators via new (p; q)-exponential function for f ∈ C[0, ∞] in (2.1) is (2.12). To get the uniform convergence and other results of approximation for Sn,p,q we suppose that sequences qn ∈ (0, pn); pn ∈ Such that qn, pn → 1 and pNn → a, qnN → b as n tending to infinity, i.e., limn→∞ 1/[n]p,q = 0.
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