Weighted polynomial and rational approximation with varying weights

In this Ph.D. thesis the problems of uniform approximation of continuous functions on closed sets of the real line by weighted polynomials {dollar}wsp{lcub}n{rcub}Psb{lcub}n{rcub}{dollar} and weighted rational functions {dollar}wsp{lcub}n{rcub}rsb{lcub}n{rcub}{dollar} with varying weight {dollar}wsp{lcub}n{rcub}{dollar} are investigated. In this type of approximation the weight varies together with the degree of the polynomial or the rational function. Recently a lot of attention has been devoted to the problem of characterizing classes of functions that are uniform limits of sequences of weighted polynomials or rational functions. The need for this type of approximation arises in several different problems on asymptotics of orthogonal polynomials, multipoint Pade approximation, and related areas. Results for some particular weights as well as general theorems including a Stone-Weierstrass type theorem for weighted polynomial approximation are known. There are only a few known results concerning weighted rational approximation. The dissertation contains several new theorems in this area.; Chapter 1 contains a brief introduction to the problem of weighted polynomial and rational approximation with varying weights. Some basic results concerning the weighted energy problem on the real line for admissible and weakly-admissible weights are presented, as well as an introduction to weighted potentials with external fields.; In Chapter 2 we investigate uniform approximation of continuous functions on unbounded sets on the real line by weighted polynomials {dollar}wsp{lcub}n{rcub}Psb{lcub}n{rcub}{dollar} with weakly-admissible weights w. The approximation at infinity depends on how dense the equilibrium measure (which in this case has noncompact support) is around that point. We show that approximation at infinity is possible if the density of the extremal measure is of the form {dollar}v(t)/tsp2{dollar} with a continuous and positive function v around infinity.; In Chapter 3 we study the approximation problem on a closed and regular subset {dollar}sum{dollar} of the real line by weighted polynomials {dollar}wsp{lcub}n{rcub}Psb{lcub}n{rcub}{dollar} when the extremal density has finitely many singularities of logarithmic type. We show that any continuous function f on {dollar}sum{dollar} that vanishes outside the set where the extremal density is positive and continuous or has a logarithmic singularity, is the uniform limit on {dollar}sum{dollar} of weighted polynomials {dollar}wsp{lcub}n{rcub}Psb{lcub}n{rcub}.{dollar} This extends previous results for continuous densities to densities having logarithmic singularities and solves an open problem posed by V. Totik.; In Chapter 4 we first consider two problems concerning uniform approximation by weighted rational functions of the form {dollar}wsp{lcub}n{rcub}rsb{lcub}n{rcub}{dollar} for the weights {dollar}w(x)=esp{lcub}x{rcub}{dollar} and {dollar}w(x)=xsp{lcub}theta{rcub},{dollar} where {dollar}rsb{lcub}n{rcub}=psb{lcub}n{rcub}/qsb{lcub}n{rcub}{dollar} is a rational function with {dollar}psb{lcub}n{rcub}{dollar} and {dollar}qsb{lcub}n{rcub}{dollar} polynomials of degree at most {dollar}lbrackalpha nrbrack{dollar} and {dollar}lbrackbeta nrbrack{dollar} respectively, for given {dollar}alpha>0{dollar} and {dollar}betageq0.{dollar} Then we prove a theorem on weighted rational approximation for a general class of weights. A necessary and a sufficient condition for weighted rational approximation with varying weights is obtained.