Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

S. Thangavelu. "Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces." Colloquium Mathematicae 94.2 (2002): 263-280. <http://eudml.org/doc/284525>.

@article{S2002,
abstract = {Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $\{Y_\{δ,j\}: δ ∈ K̂₀, 1 ≤ j ≤ d_\{δ\}\}$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_\{t\}$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_\{δ\}(iλ + ϱ)$ be the Kostant polynomials. We establish the following version of Hardy’s theorem for the Helgason Fourier transform: Let f be a function on G/K which satisfies $|f(ka_\{r\})| ≤ Ch_\{t\}(r)$. Further assume that for every δ and j the functions $F_\{δ,j\}(λ) = Q_\{δ\}(iλ +ϱ)^\{-1\} ∫_\{K/M\} f̃(λ,b)Y_\{δ,j\}(b)db$ satisfy the estimates $|F_\{δ,j\}(λ)| ≤ C_\{δ,j\}e^\{-tλ²\}$ for λ ∈ ℝ. Then f is a constant multiple of the heat kernel $h_\{t\}$.},
author = {S. Thangavelu},
journal = {Colloquium Mathematicae},
keywords = {semisimple Lie groups; symmetric spaces; Fourier transform; spherical harmonics; Jacobi functions; heat kernel},
language = {eng},
number = {2},
pages = {263-280},
title = {Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces},
url = {http://eudml.org/doc/284525},
volume = {94},
year = {2002},
}

TY - JOUR
AU - S. Thangavelu
TI - Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces
JO - Colloquium Mathematicae
PY - 2002
VL - 94
IS - 2
SP - 263
EP - 280
AB - Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis ${Y_{δ,j}: δ ∈ K̂₀, 1 ≤ j ≤ d_{δ}}$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_{t}$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{δ}(iλ + ϱ)$ be the Kostant polynomials. We establish the following version of Hardy’s theorem for the Helgason Fourier transform: Let f be a function on G/K which satisfies $|f(ka_{r})| ≤ Ch_{t}(r)$. Further assume that for every δ and j the functions $F_{δ,j}(λ) = Q_{δ}(iλ +ϱ)^{-1} ∫_{K/M} f̃(λ,b)Y_{δ,j}(b)db$ satisfy the estimates $|F_{δ,j}(λ)| ≤ C_{δ,j}e^{-tλ²}$ for λ ∈ ℝ. Then f is a constant multiple of the heat kernel $h_{t}$.
LA - eng
KW - semisimple Lie groups; symmetric spaces; Fourier transform; spherical harmonics; Jacobi functions; heat kernel
UR - http://eudml.org/doc/284525
ER -