an:02101140
Zbl 1049.43004
Andersen, Nils Byrial
Real Paley-Wiener theorems for the inverse Fourier transform on a Riemannian symmetric space
EN
Pac. J. Math. 213, No. 1, 1-13 (2004).
00107516
2004
j
43A85 22E46
Paley-Wiener theorem; Riemannian symmetric space; Laplace-Beltrami operator
Let \(G\) be a noncompact semisimple Lie group and \(K\) a maximal compact subgroup of \textit{G. S. Helgason} [Geometric analysis on symmetric spaces (Mathematical Surveys and Monographs 39, Am. Math. Soc., Providence, Rhode Island) (1994; Zbl 0809.53057)] gave a Paley-Wiener theorem for the Fourier transform, which characterizes the image of the compactly supported, infinitely differentiable functions on \(X = G/K\) in terms of holomorphic extensions and their growth at infinity, analogous to the classical case. The question addressed by the present paper is what type of Paley-Wiener type theorem is available for the inverse Fourier transform?
When restricted to \(K\)-invariant functions, the Fourier transform on \(X\) reduces to the spherical transform on \(G\). For the complex, rank one case, \textit{A. Pasquale } [Pac. J. Math. 193, No.~1, 143--176 (2000; Zbl 1014.22010)] has proved a Paley-Wiener theorem for the inverse spherical transform. The present author proves a real Paley-Wiener theorem for the inverse Fourier transform for general Riemannian symmetric spaces. Briefly, it is that for smooth \(f\) in \(L^2(X)\) to have compactly supported Fourier transform it must satisfy \(\lim\| \Delta^n f\|_2^{1/2n} < \infty\) where \(\Delta\) is the Laplace-Beltrami operator.
Benjamin B. Wells jr. (Charlotte)
Zbl 0809.53057; Zbl 1014.22010