| This dissertation consists of three parts. In the first part, we examine in detail the generalized Weierstrass problem for the generalized Jacobi weights,{dollar}{dollar}w(x) = {lcub}prodlimitssbsp{lcub}k=0{rcub}{lcub}m{rcub}{rcub} vert x - xsb{lcub}k{rcub}vertsp{lcub}thetasb k{rcub},{dollar}{dollar} {dollar}thetasb{lcub}k{rcub} geq 0, {lcub}-{rcub}1 = xsb0 < xsb1 0,{dollar} {dollar}thetasb2 > 0, x in lbrack {lcub}-{rcub}1,1rbrack.{dollar} An extension of the generalized Weierstrass problem associated with Jacobi weights to {dollar}Lsb{lcub}p{rcub}(0 < p < infty){dollar} is also investigated. For certain classes of weights, such as analytic weights, the related problems are studied.; In the second part of this dissertation we determine the upper bounds on the growth in modulus of constrained polynomials with many zeros in two disjoint intervals. M. A. Lachance and E. B. Saff investigated the polynomials of the following form {dollar}{dollar}Psb{lcub}n{rcub}(x) = {lcub}prodlimitssbsp{lcub}i=1{rcub}{lcub}s{rcub}{rcub}(x - xsb{lcub}i{rcub}){lcub}sumlimitssbsp{lcub}k=0{rcub}{lcub}m{rcub}{rcub} alphasb{lcub}k{rcub}xsp{lcub}k{rcub},{dollar}{dollar}where {dollar}s geq theta(m + s) > 0, theta > 0,{dollar} with {dollar}s{dollar} zeros in a single interval {dollar}lbrack {lcub}-{rcub}1,crbrack{dollar} and found sharp bounds for {dollar}vert Psb{lcub}n{rcub}(x)vert{dollar}. We shall study polynomials of the following more general form{dollar}{dollar}Psb{lcub}n{rcub}(x) = {lcub}prodlimitssbsp{lcub}i=1{rcub}{lcub}ssb1{rcub}{rcub} left(x - xsbsp{lcub}i{rcub}{lcub}(ssb1){rcub}right) {lcub}prodlimitssbsp{lcub}j=1{rcub}{lcub}ssb2{rcub}{rcub} left(x - xsbsp{lcub}j{rcub}{lcub}(ssb2){rcub}right) {lcub}sumlimitssbsp{lcub}k=0{rcub}{lcub}m{rcub}{rcub} alphasb{lcub}k{rcub}xsp{lcub}k{rcub},{dollar}{dollar}where {dollar}ssb1 geq thetasb1(ssb1 + ssb2 + m) > 0, ssb2 geq thetasb2(ssb1{dollar} + {dollar}ssb2 + m) > 0, thetasb1 > 0, thetasb2 > 0,{dollar} and having {dollar}ssb1{dollar}, {dollar}ssb2{dollar} zeros constrained to lie in two disjoint intervals {dollar}lbrack d,1rbrack{dollar} and {dollar}lbrack{lcub}-{rcub}1,crbrack,{dollar} respectively.; Finally, we study the limiting behavior of the zeros of Faber polynomials. We shall apply a recent result of Mhaskar and Saff on the zero distribution of asymptotically extremal polynomials to study the asymptotic distribution of zeros of Faber polynomials {dollar}Fsb{lcub}n{rcub}(z){dollar}. Three nontrivial examples of Faber polynomials associated with a regular {dollar}m{dollar}-star, an {dollar}m{dollar}-leaf lemniscate and an {dollar}m{dollar}-cusped hypocycloid are investigated in detail. The explicit formulae for the distributions of the zeros of Faber polynomials are obtained. Numerical results for the zeros of Faber polynomials of a regular {dollar}m{dollar}-star, an {dollar}m{dollar}-leaf lemniscate, an {dollar}m{dollar}-cusped hypocycloid and a regular {dollar}m{dollar}-polygon are presented. |
