Approximation in composition Real Functions Spaces

Abstract

An approximation in space of continuous functions, as we well known had studied in many books and researches, but approximation in composition of real continuous functions space had not studied previously. In these papers we studied in detail an approximation in this space, where we studied this approximation in terms of existence, uniqueness, and its degree of an element (function) of best approximation of real function that consisting from composition of two real continuous functions. We also noticed that the extent of the impact that make submission two real functions to laws of approximation in this space on the approximation of the composite function from them and degree of that approximation. After that, we studied the relationship between the module of smoothness for the original functions and the resulting function from their composition in different forms of this study and deducing their important relations for them, in the end we studied the possibility of the compact set to providing an element of best approximates of the composite function of two real functions depending on its provision these elements for the original functions which belong in it, then we concluded that this set provides a best approximation element of the function in question also vice versa for original functions.