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.
Brandolini, L., Colzani, L. (2000). Decay of Fourier transforms and summability of eigenfunction expansions. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 29(3), 611-638.
Decay of Fourier transforms and summability of eigenfunction expansions
COLZANI, LEONARDO
2000
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.