The Faber polynomials for annular sectors

Abstract

Abstract. A conformai mapping of the exterior of the unit circle to the exterior of a region of the complex plane determines the Faber polynomials for that region. These polynomials are of interest in providing near-optimal polynomial approximations in a variety of contexts, including the construction of semiiterative methods for linear equations. The relevant conformai map for an annular sector {z : R < \z\ < 1, 0 < |argz| < it), with 0 < 9 < it, is derived here and a recurrence relation is established for the coefficients of its Laurent expansion about the point at infinity. The recursive evaluation of scaled Faber polynomials is formulated in such a way that an algebraic manipulation package may be used to generate explicit expressions for their coefficients, in terms of two parameters which are determined by the interior angle of the annular sector and the ratio of its radii. Properties of the coefficients of the scaled Faber polynomials are established, and those for polynomials of degree < 15 are tabulated in a Supplement at the end of this issue. A simple closed form is obtained for the coefficients of the Faber series for 1/z . Known results for an interval, a circular arc, and a circular sector are reproduced as special cases.

The conformal ‘bratwurst’ mapsand associated Faber polynomials

“…Often the boundary of Ω is not chosen to be analytic and in the computation of Ψ , the Schwarz-Christoffel mapping formula is employed. Particular examples of the use of this formula are the hybrid Arnoldi-Faber method [32], the adaptive (k, l)-step method by Li [19] as well as the development by Coleman and Myers [1], who compute a conformal map and Faber polynomials for annular sectors. A disadvantage of the Schwarz-Christoffel approach is that the resulting formulas include quantities which are determined implicitly through integral equations.…”

“…The second class consists of non-convex sets with a more complicated geometry and non-analytic boundary, for example annular sectors [1,24] and (non-convex) sets bounded by polygons [21,32]. But while the nonconvexity of these sets leads to broader applicability, the non-analytic boundary makes the computation of the exterior mapping function difficult.…”

“…We have indicated these difficulties in the introduction. 1 We remark that the definition of the boundary rotation of a compact set Ω used by Starke and Varga (see [32,Formula (3.6)]) is not the same as the classical definition as given in [27, p. 5] (also cf. [18, p. 197]).…”

Expansion of analytic functions of an operator in series of Faber polynomials

“…The fact that the curve F i + e is bounded away from Sp(T) and standard continuity (the curve Fi + E is analytic) and compactness arguments, together with Walsh's theorem, yield sup (|p n ( 7 (t))| ||(7(*K-r)" 1 …”

Accelerated polynomial approximation of finite order entire functions by growth reduction

“…An efficient method for the numerical evaluation of T n is described in [5]. Moreover, in [4] and [10] explicit expressions for the Faber polynomials F n,K in the cases of K being a circular or an annular sector are given.…”