In this article, we introduce generalized q−Sz´asz-Mirakjan operators and study their approximation properties. Based on the Voronovskaja’s theorem, we obtain quantitative estimates for these operators.
A topological space X is called P-normal if there exist a normal space Y and a bijective function f : X −→ Y such that the restriction f|A : A −→ f(A) is a homeomorphism for each paracompact subspace A ⊆ X. We will investigate this property and produce some examples to illustrate the relation between P-normality and other weaker kinds of normality.
We study the problem of dichotomy and boundedness for impulsive dynamic equations on arbitrary closed subset of real numbers. The spectral decomposition theorem gives all our main results. The obtained results are fundamentally new, even for the classical case.
We give generalizations and refinements of Jensen and Jensen− Mercer inequalities by using weights which satisfy the conditions of Jensen and Jensen− Steffensen inequalities. We also give some refinements for discrete and integral version of generalized Jensen−Mercer inequality and shown to be an improvement of the upper bound for the Jensen’s difference given in [32]. Applications of our work include new bounds for some important inequalities used in information theory, and generalizing the relations among means.
APPROXIMATION BY GENERALIZED q-SZASZ-MIRAKJAN ´ OPERATORS
