Topics in approximation theory

As the first result of this dissertation, we determine, in Chapter II, the functions on ({dollar}-{dollar}1,1) that are uniform limits of weighted polynomials of the form (1 {dollar}-{dollar} {dollar}x)sp{lcub}alphasb{lcub}n{rcub}{rcub}(1 + x)sp{lcub}betasb{lcub}n{rcub}{rcub}psb{lcub}n{rcub}(x){dollar}, where deg {dollar}psb{lcub}n{rcub}{dollar} {dollar}leq{dollar} {dollar}n{dollar}, {dollar}limlimitssb{lcub}ntoinfty{rcub} {lcub}alphasb{lcub}n{rcub}over n{rcub}{dollar} = {dollar}thetasb1{dollar} {dollar}geq{dollar} 0 and {dollar}limlimitssb{lcub}ntoinfty{rcub} {lcub}betasb{lcub}n{rcub}over n{rcub}{dollar} = {dollar}thetasb2{dollar} {dollar}geq{dollar} 0. Estimates for the rate of convergence are also obtained. Our results confirm a conjecture of E. B. Saff for the case when {dollar}alphasb{lcub}n{rcub}{dollar} = {dollar}nthetasb1{dollar} {dollar}>{dollar} 0 and {dollar}betasb{lcub}n{rcub}{dollar} = {dollar}nthetasb2{dollar} {dollar}>{dollar} 0, and extend previous results of G. G. Lorentz, M. v. Golitschek, E. B. Saff and R. Varga for incomplete polynomials.; Another aspect of our research concerns a comparison of the convergence rate of a Fourier series on the whole interval versus its convergence rate on a subinterval. For a function {dollar}f{dollar} defined on the interval {dollar}I {lcub}:{rcub}{lcub}={rcub}{dollar} ({dollar}-{dollar}1,1), let {dollar}psbsp{lcub}n,2{rcub}{lcub}*{rcub}(f){dollar} be the polynomial of degree at most {dollar}n{dollar} of best approximation to {dollar}f{dollar} with respect to the {dollar}Lsb2{dollar} norm{dollar}{dollar}Vert gVertsb{lcub}Lsb2(dalpha){rcub} {lcub}:{rcub}{lcub}={rcub} left(intsb{lcub}I{rcub}vert g(x)vertsp2dalpharight)sp{lcub}1/2{rcub},{dollar}{dollar}where {dollar}dalpha{dollar} is a finite, positive Borel measure on {dollar}I{dollar}. We show that if {dollar}alphaspprime{dollar} {dollar}>{dollar} 0 a.e. on {dollar}I{dollar}, then for {dollar}f{dollar} {dollar}in{dollar} {dollar}Lsb2(dalpha){dollar}, {dollar}f{dollar} not a polynomial, and {dollar}-{dollar}1 {dollar}leq{dollar} {dollar}a{dollar} {dollar}<{dollar} {dollar}b{dollar} {dollar}leq{dollar} 1, we have{dollar}{dollar}sumlimitssbsp{lcub}n=0{rcub}{lcub}infty{rcub}left({lcub}Vert f-psbsp{lcub}n,2{rcub}{lcub}*{rcub}(f)Vertsb{lcub}Lsb2(dalpha,lbrack a,brbrack){rcub}overVert f-psbsp{lcub}n,2{rcub}{lcub}*{rcub}(f)Vertsb{lcub}Lsb2(dalpha){rcub}{rcub}right)sp2 = inftyeqno(1){dollar}{dollar}where {dollar}Vert gVertsb{lcub}Lsb2(dalpha,lbrack a,brbrack){rcub} {lcub}:{rcub}{lcub}={rcub} (intsbsp{lcub}a{rcub}{lcub}b{rcub}vert g(x)vertsp2dalpha)sp{lcub}1/2{rcub}{dollar}. Furthermore, in (1) the exponent 2 is sharp in the sense that it cannot be replaced by any larger number. Roughly speaking, (1) means that {dollar}{lcub}psbsp{lcub}n,2{rcub}{lcub}*{rcub}(f){rcub}sbsp{lcub}n=0{rcub}{lcub}infty{rcub}{dollar} does not approximate {dollar}f{dollar} substantially better on any subinterval of {dollar}I{dollar} than it does on the whole interval (there can be an improvement of at most a factor of O({dollar}{lcub}1oversqrt{lcub}n{rcub}{rcub}{dollar})). We also investigate the distribution of zeros of the {dollar}Lsb{lcub}p{rcub}{dollar} ({dollar}p{dollar} {dollar}>{dollar} 0) best approximants, and generalize classical Jentzsch-Szego type theorems. In so doing, we obtain a result about the regularity of measures (cf. Definition 3.1 in Chapter III). Corresponding results are also obtained for approximation on the unit circle {dollar}{lcub}z{dollar} {dollar}in{dollar} C: {dollar}vert zvert{dollar} = 1{dollar}{rcub}{dollar}. Finally, in Chapter IV an analogue of (1) is obtained for the {dollar}Lsb2{dollar} approximation on the whole real line R with respect to Freud weights. All of these results support the principle of contamination introduced by Dr. Saff (cf. Chapter III, S1).