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An Algebraic Theorem of E. Fischer, and the Holomorphic Goursat Problem

An Algebraic Theorem of E. Fischer, and the Holomorphic Goursat Problem AN ALGEBRAIC THEOREM OF E. FISCHER, AND THE HOLOMORPHIC GOURSAT PROBLEM HAROL D S. SHAPIRO Dedicated, in friendship, to Donald J. Newman 1. Introduction 1.1. Background. This paper is a somewhat amplified version of two lectures presented at the 'Summer Conference on Potential Theory' in 1983 in Durham, organized by the London Mathematical Society. My original motivation for studying this material was a theorem of Ernst Fischer which generalizes the classical fact that every homogeneous polynomial in (x ...,x ) is uniquely representable as the sum of 15 n a (polynomial) solution of Laplace's equation, and a multiple of x\ + ... + ,x ,. One line of generalization of Fischer's Theorem was pursued in several joint papers with D. J. Newman [25, 26, 27] and led to the study of the Hilbert space of entire func- tions of (z ...,z ) that are square integrable with respect to the weight function 15 n 2 2 exp( — (IzJ -!-... + |z | )). The present paper pursues a different kind of generalization of Fischer's Theorem, which relates it to two central areas of partial differential equations: (i) partial differential equations in the complex domain, especially the Cauchy and http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

An Algebraic Theorem of E. Fischer, and the Holomorphic Goursat Problem

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/21.6.513
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Abstract

AN ALGEBRAIC THEOREM OF E. FISCHER, AND THE HOLOMORPHIC GOURSAT PROBLEM HAROL D S. SHAPIRO Dedicated, in friendship, to Donald J. Newman 1. Introduction 1.1. Background. This paper is a somewhat amplified version of two lectures presented at the 'Summer Conference on Potential Theory' in 1983 in Durham, organized by the London Mathematical Society. My original motivation for studying this material was a theorem of Ernst Fischer which generalizes the classical fact that every homogeneous polynomial in (x ...,x ) is uniquely representable as the sum of 15 n a (polynomial) solution of Laplace's equation, and a multiple of x\ + ... + ,x ,. One line of generalization of Fischer's Theorem was pursued in several joint papers with D. J. Newman [25, 26, 27] and led to the study of the Hilbert space of entire func- tions of (z ...,z ) that are square integrable with respect to the weight function 15 n 2 2 exp( — (IzJ -!-... + |z | )). The present paper pursues a different kind of generalization of Fischer's Theorem, which relates it to two central areas of partial differential equations: (i) partial differential equations in the complex domain, especially the Cauchy and

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1989

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