Inequalities for generalized polynomials and their applications.

摘要

Let(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}rm f(z) = prodlimitssbsp{lcub}j=1{rcub}{lcub}k{rcub}vert z-zsb{lcub}j{rcub}vertsp{lcub}rsb j{rcub}qquad (zsb{lcub}j{rcub}in C, rsb{lcub}j{rcub}>0 are real).leqno(1){dollar}{dollar}(TABLE/EQUATION ENDS)The function f is called the modulus of a generalized complex algebraic polynomial of degree(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}rm N = sumlimitssbsp{lcub}j=1{rcub}{lcub}k{rcub}rsb{lcub}j{rcub}.leqno(2){dollar}{dollar}(TABLE/EQUATION ENDS)In the trigonometric case we say that(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}rm f(z) = prodlimitssbsp{lcub}j=1{rcub}{lcub}k{rcub}vertsin((z-zsb{lcub}j{rcub})/2)vertsp{lcub}rsb j{rcub}qquad (zsb{lcub}j{rcub}in C, rsb{lcub}j{rcub}>0 are real)leqno(3){dollar}{dollar}(TABLE/EQUATION ENDS)is the modulus of a generalized complex trigonometric polynomial of degree(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}rm N = {lcub}1over2{rcub} sumlimitssbsp{lcub}j=1{rcub}{lcub}k{rcub}rsb{lcub}j{rcub}.leqno(4){dollar}{dollar}(TABLE/EQUATION ENDS); A number of important inequalities holding for ordinary polynomials can be extended for generalized polynomials by writing generalized degree in place of the ordinary one. Usually such an extension is far from being straightforward. We have obtained sharp Markov, Bernstein, Schur, Nikolskii and Remez type inequalities for generalized polynomials in both the algebraic and trigonometric cases. Some results (e.g. the trigonometric and pointwise algebraic Remez-type inequalities) are new even for ordinary polynomials. As an application we give sharp upper bounds for the consecutive zeros of orthogonal polynomials associated with weightfunctions from rather wide families, far beyond the well-known Szego class. The Thesis consists of six research papers (see the Contents) on the subject. The reader will find an abstract and an introduction in the beginning of each paper. Though the Remez-type inequalities proved in the first paper play a significant role in the whole Thesis, by accepting the results from there, the papers may be read independently of each other. Further results and applications are expected in approximation theory, numerical analysis, the theory of orthogonal polynomials and potential theory as well.