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Instability Phenomena for the Fourier Coefficients
Author(s) -
Aizenberg Lev,
Tarkhanov Nikolai
Publication year - 1996
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19961820102
Subject(s) - mathematics , orthogonality , sobolev space , fourier transform , mathematical analysis , pure mathematics , open set , distribution (mathematics) , operator (biology) , combinatorics , geometry , biochemistry , chemistry , repressor , transcription factor , gene
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 D and σ in X with σ ⊂⊂ D ⊂⊂ X , we fix a sequence { e v } of solutions of Pu = 0 in D which are pairwise orthogonal under integration over both D and σ. By orthogonality is meant the orthogonality in the corresponding Sobolev spaces; we also assume a completeness of the system on σ. For a fixed y ε X/⊂, denote by k v (y) the Fourier coefficients of a fundamental solution ϕ(·, y ) of P with respect to the restriction of {e v } to σ. Suppose K is a compact set in D/σ , and let f be a distribution with support on K . In this paper we show, under appropriate conditions on K , that if the moments ( f, 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 of f .