Some Aspects of Approximation Theory Based on Various Positive Linear Operators
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Description
Approximation theory is a wide area of research in the field of mathematics. This work mainly focuses on the convergence results for linear positive operators. After Weierstrass’ well known approximation theorem and Korovkin’s significant convergence theorem, many operators were proposed and constructed by various mathematicians. We study simultaneous approximation for the linear combinations of modified Beta operators and obtain an error estimate and rate of approximation in terms of higher order modulus of continuity. In this book, the author presents direct, inverse and saturation theorems for modified Beta operators. Direct estimates in both local and global approximation are also discussed. Moments generation is also presented for the operators in each chapter. Asymptotic expansion of linear positive operators, which is important for convergence estimates, is also discussed. By considering the linear combinations, we slacken the positivity conditions of the operators. The author discusses linear and iterative combinations and presents some results. Stancu type generalization of Baskakov and Szasz basis functions is also presented. The author mentions approximation properties of linear positive operators using quantum calculus and post quantum calculus. The author also proposes a Stancu type generalization of the Phillips operators. Towards the end, convergence estimates for Post-Widder operators in complex plane is discussed. We discuss the generalized modified Srivastava-Gupta operators by using iterative combinations in ordinary and simultaneous approximation.