The Faber polynomials for circular sectors

Abstract:

The Faber polynomials for a region of the complex plane, which are of interest as a basis for polynomial approximation of analytic functions, are determined by a conformal mapping of the complement of that region to the complement of the unit disc. We derive this conformal mapping for a circular sector $\{ {z:|z|\; \leqslant 1,\;|\arg z|\; \leqslant \pi /\alpha } \}$, where $\alpha > 1$, and obtain a recurrence relation for the coefficients of its Laurent expansion about the point at infinity. We discuss the computation of the coefficients of the Faber polynomials of degree 1 to 15, which are tabulated here for sectors of half-angle ${5^ \circ }$, ${10^\circ }$, ${15^ \circ }$, ${30^ \circ }$, ${45^\circ }$, and ${90^ \circ }$, and we give explicit expressions, in terms of $\alpha$, for the polynomials of degree $\leqslant 3$. The norms of Faber polynomials are tabulated and are compared with those of the Chebyshev polynomials for the same regions.