| First, in Chapter I, we give the outline of this dissertation. In Chapter II, we investigate a question of Price and Morris: When is an interpolating projection a minimal one? In particular, we construct a two-dimensional subspace {dollar}V subset C(K){dollar} such that an interpolating projection on {dollar}V{dollar} is a minimal projection with the norm {dollar}>{dollar}1. This answers a question posed by B. L. Chalmers as well as a question asked implicitly in a theorem of E. W. Cheney and P. Morris. We also give a quantitative generalization of the above mentioned theorem. Trace-duality is used to obtain these results.; Another aspect of our research concerns asymptotics of {dollar}Lsb{lcub}p{rcub}{dollar} extremal polynomials on the unit circle. Let {dollar}p > 1{dollar} and {dollar}dmu{dollar} be a positive finite Borel measure on the unit circle {dollar}Gamma{dollar}:= {dollar}{lcub}z in {lcub}bf C{rcub} : vert zvert = 1{rcub}.{dollar} Define the monic polynomial {dollar}phisb{lcub}n,p{rcub}(z) = zsp{lcub}n{rcub} + cdots in {lcub}cal P{rcub}sb{lcub}n{rcub}{dollar} (the set of polynomials of degree at most {dollar}n{dollar}) by the extremal property{dollar}{dollar}intsbGammavertphisb{lcub}n,p{rcub}(z)vertsp{lcub}p{rcub}dmu = {lcub}inflimitssb{lcub}Pin{lcub}cal P{rcub}sb{lcub}n-1{rcub}{rcub}{rcub} intsbGammavert zsp{lcub}n{rcub} + Pvertsp{lcub}p{rcub} dmu.{dollar}{dollar}Under certain conditions on {dollar}dmu{dollar}, the asymptotics of {dollar}phisb{lcub}n,p{rcub}(z){dollar} are obtained which yield generalizations of Szego's theorem for {dollar}p ne 2.{dollar} The results are for {dollar}z{dollar} outside as well as on {dollar}Gamma{dollar}. Zero distributions of {dollar}phisb{lcub}n,p{rcub}(z){dollar} are also discussed, which generalize theorems of Nevai and Totik, Mhaskar and Saff for {dollar}p ne 2.{dollar}; Finally, in Chapter IV, we study best polynomial approximants with linear constraints. Let {dollar}A{dollar} be a {dollar}(k + 1) times (k + 1){dollar} matrix. For {dollar}p in {lcub}cal P{rcub}sb{lcub}n{rcub},{dollar} denote {dollar}underline{lcub}p{rcub}{dollar}:= {dollar}(p(0),pprime(0),...,psp{lcub}(k){rcub}(0))sp{lcub}T{rcub}{dollar} and {dollar}Bsb{lcub}n{rcub}(A){dollar}:= {dollar}{lcub}p in {lcub}cal P{rcub}sb{lcub}n{rcub} : Aunderline{lcub}p{rcub} = underline{lcub}0{rcub}{rcub}.{dollar} Let {dollar}E subset {lcub}bf C{rcub}{dollar} be a compact set that does not separate the plane and {dollar}f{dollar} be a function continuous on {dollar}E{dollar} and analytic in the interior of {dollar}E{dollar}. Set(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}eqalign{lcub}Esb{lcub}n{rcub}(A,f) {lcub}:={rcub} inf{lcub}Vert f - pVertsb{lcub}E{rcub}&: p in Bsb{lcub}n{rcub}(A){rcub}cr Esb{lcub}n{rcub}(f) {lcub}:={rcub} inf{lcub}Vert f - pVertsb{lcub}E{rcub}&: p in {lcub}cal P{rcub}sb{lcub}n{rcub}{rcub}.cr{rcub}leqnorm and{dollar}{dollar}(TABLE/EQUATION ENDS)Our goal is to study approximation to {dollar}f{dollar} on {dollar}E{dollar} by polynomials from {dollar}Bsb{lcub}n{rcub}(A).{dollar} We obtain necessary and sufficient conditions on the matrix {dollar}A{dollar} for the convergence {dollar}Esb{lcub}n{rcub}(A,f) to 0{dollar} to take place. These results depend on whether zero lies inside, on or outside {dollar}E{dollar} and yield generalizations of a theorem of Clunie, Hasson and Saff for approximation by polynomials that omit a power of {dollar}z{dollar}. Let {dollar}Psbsp{lcub}n,k{rcub}{lcub}*{rcub} in Bsb{lcub}n{rcub}(A){dollar} be such that {dollar}Esb{lcub}n{rcub}(A,f) = Vert f - Psbsp{lcub}n,k{rcub}{lcub}*{rcub}Vert.{dollar} We also study the asymptotic behavior of the zeros of {dollar}Psbsp{lcub}n,k{rcub}{lcub}*{rcub}{dollar} and the relations between {dollar}Esb{lcub}n{rcub}(f){dollar} and {dollar}Esb{lcub}n{rcub}(A,f).{dollar}... |
