PROCEEDINGS of the AMERICANMATHEMATICAL SOCIETY Volume 111, Number 3, March 1991 A SAMPLING THEOREM FOR A CLASS OF PSEUDOANALYTIC FUNCTIONS J. L. SCHIFF AND W. J. WALKER (Communicated by J. Marshall Ash) Abstract. The //-regular class of pseudoanalytic functions satisfy the Cauchy- Riemann equations for Au = p u. A sampling algorithm is given which ex- presses the Fourier coefficients of these functions as a countable sum of sample values taken around a circle. This representation is obtained using Möbius inversion. 1. Introduction The potential of the strong nuclear force can be described by the solution e'^/r of the elliptic equation (1) Au = p u (p > 0). This description was proposed by the Japanese physicist Hideki Yukawa, and equation (1) now bears his name. Subsequently, Duffin [2] undertook the de- velopment of a Yukawan potential theory, which forms the basis of our present work. As /i->0,a solution of the Yukawa equation becomes a solution of the Laplace equation. In this article, attention is restricted to the two-dimensional case of (1). Functions which are C -solutions of ( 1) in a domain of the complex plane are termed panharmonic. They give rise to pseudoanalytic functions f = u + iv , where u and v are panharmonic and satisfy the Cauchy-Riemann equations for (1). These pseudoanalytic functions are termed p-regular, and have an el- egant Fourier series representation, the coefficients of which seem to satisfy a new Bieberbach condition (cf. [5]). Our aim is to give a sampling theorem. In our case it yields an exact rep- resentation of the Fourier coefficients of a /¿-regular function / by taking a countable set of values of / on the boundary of a circle. The authors in [3] have already given a similar algorithm for the Taylor coefficients of an analytic Received by the editors October 25, 1988 and, in revised form, February 12, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 30G20; Secondary 30B99, 30E10. Key words and phrases. Möbius function, panharmonic, /¿-regular, pseudoanalytic, sampling theorem. ©1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page 695 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use