Instability Phenomena for the Fourier Coefficients

✍ Scribed by Lev Aizenberg; Nikolai Tarkhanov

• Publisher: John Wiley and Sons
• Year: 1996
• Tongue: English
• Weight: 732 KB
• Volume: 182
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19961820102

✦ Synopsis

Let P be an elliptic differential operator on a non-compact connected manifold X ; suppose that both X and the coefficients of P are real analytic. Given a pair of open sets V and u in X with cr CC V C C X , we fix a sequence { e v } of solutions of Pu = 0 in V which are pairwise orthogonal under integration over both V and 0. By orthogonality is meant the orthogonality in the corresponding Sobolev spaces; we also assume a completeness of the system on u . For a fixed y E X \F, denote by k,(y) the Fourier coefficients of a fundamental solution a( , y) of P with respect to the restriction of {e,} to u. Suppose K is a compact set in V \ (T, and let f be a distribution with support on K . In this paper we show, under appropriate conditions on K , that if the moments ( j , k v ) decrease sufficiently rapidly in a certain precise sense, then these moments vanish identically. In the most favorable cases, it is then possible to conclude that f = 0. This phenomenon was previously noticed by the first author and L. ZALCMAN for analytic and harmonic moments o f f