A local Paley–Wiener theorem for compact symmetric spaces

Abstract

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M = U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.

Fourier Series on Compact Symmetric Spaces: K-Finite Functions of Small Support

“…It is known from [19], Theorem 4.2(i), that the estimate in Property (b) holds for K-invariant functions on U/K. We prove the property in general by reduction to that case.…”

“…We recall some basic notation from [19]. We are considering a Riemannian symmetric space U/K, where U is a connected compact semisimple Lie group and K a closed symmetric subgroup.…”

“…We recall that if U is simply connected, then U θ is connected and K = K 0 , see [13], p. 320. We recall also that in general K = MK 0 , see [19], Lemma 5.2.…”

“…Notice that the Paley-Wiener space PW r (a) defined in [19] can be identified with the space of functions ϕ in PW K,r (a), for which ϕ(λ) is a constant function on K/M for each λ. This follows from the normalization (4.4).…”

“…Proof The relevant value of R is the same as in [19], Theorem 4.2(iii) and Remark 4.3. The lemma follows easily from application of [19], Sect.…”