Contributions to the theory of approximation by polynomials

In this dissertation we study several problems of approximation to a continuous function by trigonometric and algebraic polynomials in the uniform norm and {dollar}Lsb p{dollar} norm. The first chapter contains a brief discussion of the problems considered.; In the second chapter we consider approximation by incomplete polynomials. We prove a Kadec type result concerning the distribution of the extreme points of the error curve of the best approximation by incomplete polynomials. It is also shown that the zeros of the constrained Chebyshev polynomials are a good choice as interpolating nodes for incomplete polynomial interpolation. The error of best approximation by incomplete polynomials is also studied and a counterexample to a conjecture of Hasson is given.; In the third chapter we discuss the convergence of the derivatives of approximating polynomials. The main theorem gives sufficient conditions for a sequence of trigonometric polynomials, approximating a given function {dollar}f(x){dollar} to have their {dollar}k{dollar}th derivatives converge to the {dollar}k{dollar}th derivative of {dollar}f(x){dollar} at a point or on an interval. Applying this to some special settings, we obtain several new and known results. Analogous results for {dollar}Lsb p{dollar} space and algebraic polynomials are also given. We also discuss the sharpness of the estimates by giving a negative theorem and several examples. In the last section of this chapter the convergence of derivatives of specific interpolating polynomials is considered.; The last chapter studies the rate of approximation on disjoint intervals, which is related to some results of Chui and Hasson.