Multi-dimensional polynomial inequalities: Norms of interpolation operators.

摘要

In this dissertation we discuss various topics in approximation theory, including multi-dimensional polynomial inequalities and extremum problems in the Hardy space, Hinfinity (D), consisting of all functions which are analytic and bounded in the open unit disk D.;We prove a two-dimensional version of Turan's first main theorem, along with some applications. Specifically, we consider the power sum S(i, j) = l=1n k=1m bklwkizl j, for bkl, wk, zl ∈ C , and provide an upper bound for |S(0, 0)| in terms of the maximum of |S| over the lattice of points of a region in R2 which is a given distance from the origin.;We also provide a multi-dimensional version of Nazarov's extension of Turan's lemma---a theorem in which the uniform norm of a complex valued polynomial p defined on the unit circle T is compared with the uniform norm of p on any measurable subset of T . If we let Tn := T x ··· x T represent the distinguished boundary of the polydisk D n := D x ··· x D for some n ∈ N then, as in the one-dimensional case, the constant which relates the uniform norm of p on Tn to the uniform norm of p on any measurable subset of Tn depends on the order, i.e., the number of non-zero coefficients, of p and the measure of the set E.;The second topic of this dissertation concerns calculating the norms of various Lagrange interpolation functionals acting on the Hardy space Hinfinity (D). For example, we calculate the norm of Ln-1(·;zeta), where Ln-1(·;zeta) represents the Lagrange interpolation polynomial of degree n - 1, evaluated at some complex number zeta, and defined by interpolating functions in Hinfinity (D) at the zeros of zn - rn, for various values of n and zeta. We assume that 0 r 1 and that |zeta| > 1. We also calculate the norm of L1(·;zeta) in the case that L1(·;zeta) is defined by interpolating functions in Hinfinity (D) at two arbitrary fixed values in (-1,1), as opposed to the symmetrical values r and - r.