A shortcut to weighted representation formulas for holomorphic functions

Abstract

As a by-product a variety of projection kernels (P such that f = f Pf, f o r f h o l o morphic) were obtained. These kernels give representation formulas for holomorphie functions which in general consist of an integral over the whole domain and a boundary integral. The projection part and the corresponding representation formulas have proved to be quite fruitful. They have been used by several authors (see e.g. [2], [3], [5] and [ 10]) to obtain explicit solutions to division and interpolation problems. The purpose of this paper is to give a short proof of a generalization of the representation formulas in [3] and [4] without making the d6tour to the 0-problem and the kernels K. We derive in w 1 a quite general formula (Theorem l) which is then turned into a more tangible one for bounded domains (Theorem 2). Using logarithmic residues we also obtain weighted versions of certain formulas in [13] and [15]. In w 2 we give a few examples and comments. To motivate what follows, let us take a brief look at the case n= I. Let f be holomorphic in a domain f 2 c C and suppose that 12CCa(OX~). We then have