Principal Functions

  • Burton Rodin
  • Leo Sario

Part of the The University Series in Higher Mathematics book series (USHM)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Burton Rodin, Leo Sario
    Pages 1-13
  3. Burton Rodin, Leo Sario
    Pages 14-37
  4. Burton Rodin, Leo Sario
    Pages 38-80
  5. Burton Rodin, Leo Sario
    Pages 81-137
  6. Burton Rodin, Leo Sario
    Pages 138-192
  7. Burton Rodin, Leo Sario
    Pages 193-210
  8. Burton Rodin, Leo Sario
    Pages 211-231
  9. Burton Rodin, Leo Sario
    Pages 232-286
  10. Burton Rodin, Leo Sario
    Pages 287-304
  11. Back Matter
    Pages 305-347

About this book


During the decade and a half that has elapsed since the intro­ duction of principal functions (Sario [8 J), they have become impor­ tant tools in an increasing number of branches of modern mathe­ matics. The purpose of the present research monograph is to systematically develop the theory of these functions and their ap­ plications on Riemann surfaces and Riemannian spaces. Apart from brief background information (see below), nothing contained in this monograph has previously appeared in any other book. The basic idea of principal functions is simple: Given a Riemann surface or a Riemannian space R, a neighborhood A of its ideal boundary, and a harmonic function s on A, the principal function problem consists in constructing a harmonic function p on all of R which imitates the behavior of s in A. Here A need not be connected, but may include neighborhoods of isolated points deleted from R. Thus we are dealing with the general problem of constructing harmonic functions with given singularities and a prescribed behavior near the ideal boundary. The function p is called the principal function corresponding to the given A, s, and the mode of imitation of s by p. The significance of principal functions is in their versatility.


Eigenvalue Hilbert space Riemann surface convergence derivative extrema function holomorphic function integral integration interpolation logarithm maximum measure operator

Authors and affiliations

  • Burton Rodin
    • 1
  • Leo Sario
    • 2
  1. 1.University of CaliforniaSan DiegoUSA
  2. 2.University of CaliforniaLos AngelesUSA

Bibliographic information