On the expansion of analytic functions in series of polynomials

Abstract

where the functions Pk(z) are properly chosen. The series (1) converges uniformly in the closed region interior to C. This fundamental result is a direct generalization of Taylor's series, to which Faber's series (1) reduces when C is a circle. On any circle C for which Taylor's selies converges, if the center of C is the point about which the Taylor development is considered, Taylor's series reduces precisely to Fourier's series, both formally and in fact. More generally, Laurent's series similarly reduces to Fourier's series and conversely, if the function considered is defined and integrable on the circle C. The natural generalization of Fourier's series and of Laurent's series to the case of an arbitrary contour C seems not to have been made. It is the object of the present paper to set forth such a generalization, as indicated by the following theorem: