Universal Series and Fundamental Solutions of the Cauchy-Riemann Operator

Abstract

We prove the existence of universal series whose terms are fundamental solutions, or derivatives of them, of the Cauchy-Riemann operator. By universal we mean a series whose partial sums are dense, with respect to the uniform topology on compacta, in the space of functions holomorphic on a certain subset of the complex plane. We show in particular that the coefficients of the series may be chosen to belong to some subspace of complex sequences like lp, 1 < p < ∞, or c 0.