Decay of Fourier transforms and summability of eigenfunction expansions

Fourier coefficients ∫01 f(x)exp (-2πinx)dx of piecewise smooth functions are of the order of |n|-1 and Fourier series ∑n=-∞+∞ f(n) Qxp(2π inx) converge everywhere. Here we consider analogs of these results for eigenfunction expansions f(x) = ∑λ, F f(λ)φλ (x) where {λ2}and {φλ(x)} are eigenvalues and an orthonormal complete system of eigenfunctions of a second order positive elliptic operator on a -dimensional manifold. We prove that the norms of projections of piecewise smooth functions on subspaces generated by eigenfunctions with Λ ≤ λ, ≤ Λ + 1 satisfy the estimates {∑Λ ≤ λ, ≤ Λ + 1 | F f (λ)|2}1/2 ≤ c Λ-1. Then we give some sharp results on the Riesz sunmiability of Fourier series. In particular we prove that the Riesz means ∑λ (N — 3)/2 converge. © Scuola Normale Superiore, Pisa, 2000, tous droits réservés.