Emphasizing on Simultaneous Approximation of Unbounded Functions

Abstract

The word approximation is derived from Latin word “approximatus”. The term can be applied in our surroundings to various properties (e.g. value, quantity, image, description) that are nearly, but not exactly correct or similar, but not exactly the same. Mostly approximation is often applied to numbers but it is also frequently applied to such things as mathematical functions, shapes, and physical laws [1,2]. In science, approximation can refer to use a simpler process or model when the correct model is difficult to use. An approximate model is used to make calculations easier. Approximations might also be used if incomplete information prevents use of exact representations. The type of approximation depends on the available information, the degree of accuracy, sensitivity of the problem to the data and the savings (usually in time and effort).

In this chapter, we present a systematic overview of approximation by the use of linear positive operators, a useful tool used to increase the order of approximation [2]. The properties of operators are not only limited to the functions of bounded variation but unbounded variation also [3]. Many authors studied and used the rate of convergence, Moduli of smoothness etc. to get various results for several operators.

Here we use summation-integral type linear positive operators to approximate an unbounded function. These operators can be named as Dual Beta type operators. We obtain moments and other type of results in simultaneous approximation and Voronovskaya type asymptotic formula for this new sequence of operators. We also find important direct theorem for these operators.