$L^2$-Approximations of power and logarithmic functions with applications to numerical conformal mapping

Abstract

For a bounded Jordan domain G with quasiconformal boundary L, two-sided estimates are obtained for the error in best L 2 (G) polynomial approximation to functions of the form (z − τ ) β , β > −1, andFurthermore, Andrievskii's lemma that provides an upper bound for the L ∞ (G) norm of a polynomial p n in terms of the L 2 (G) norm of p n is extended to the case when a finite linear combination (independent of n) of functions of the above form is added to p n . For the case when the boundary of G is piecewise analytic without cusps, the results are used to analyze the improvement in rate of convergence achieved by using augmented, rather than classical, Bieberbach polynomial approximants of the Riemann mapping function of G onto a disk. Finally, numerical results are presented that illustrate the theoretical results obtained.

Finite-Term Relations for Planar Orthogonal Polynomials

“…Orthogonal polynomials over planar domains, as well as orthogonal polynomials over planar curves (the so-called Szegö polynomials), are becoming increasingly popular. This is partly due to their relevance in polynomial approximation and reconstruction problems, see for instance [8,15,[18][19][20][21]. It does not mean that complex orthogonal polynomials were not studied before, but the amount of work and publications devoted to their onedimensional cousins (orthogonal polynomials with respect to measures supported on the real line or the circle) was until recently disproportionately large.…”

Error Analysis of the Bergman Kernel Method with Singular Basis Functions

“…Proof. The proof is based on Andrievskii's lemma for singular algebraic functions given in [12,Corollary 2.5] and relies on the results contained in [12, §2]. The details of the derivation are as follows: First, we note that our assumption implies that Γ is a quasiconformal curve.…”

“…4] that there are cases where the exponent s can not be replaced by a smaller number. However, the factor log n can be replaced by √ log n, see [1] and [12,Rem. 3.1].…”

“…provided that 1/(2 − λ) is not a positive integer, where c is a constant that does not depend on n, was established in [12,Thm. 3.2] by Maymeskul, Saff and the second author.…”

“…3.2] by Maymeskul, Saff and the second author. The theoretical justification of the BKM/AB with basis function that reflect the corner singularities of f 0 was given in [12], by means of sharp estimates for the associated BKM/AB errors. The purpose of the present paper is to derive theoretical results that justify the use of basis functions that reflect (a) pole singularities of f 0 and (b) both corner and pole singularities of f 0 .…”