A Uniqueness Theorem for Eigenfunction Expansions.

the series on the right of (3) being called the Fourier Eigenfunction Series and a. the Fourier Coefficients of f(x, y). I have studied elsewhere' the problem of convergence and summability of a Fourier Eigenfunction Series. In this note I am interested in announcing a result on uniqueness of eigenfunction expansion. Actually, we have thfe following, THEOREM. Let us suppose we are given an eigenfuzaction series ajw1{x, y) + a2w2(x, Y) + a3w3(x, y) + . ., a,, a2, ... real (5) satisfying the following properties (i) There exists a continuous function +(x, y) defined on D + C and vanishing on C such that