A Paley-Wiener theorem for the inverse Fourier transform on some homogeneous spaces

We formulate and prove a version of Paley-Wiener theorem for the inverse Fourier transform on non-compact Riemannian symmetric spaces and Heisenberg groups. The main ingredient in the proof is the Gutzmer's formula.Comment: 17 page

An analogue of Gutzmer's formula for Hermite expansions

We prove an analogue of Gutzmer's formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of $L^2(\R^n)$ under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.Comment: 15 page

Heat kernel transform for nilmanifolds associated to the Heisenberg group

We study the heat kernel transform on a nilmanifold $M$ of the Heisenberg group. We show that the image of $L^2(M)$ under this transform is a direct sum of weighted Bergman spaces which are related to twisted Bergman and Hermite-Bergman spaces.Comment: Revised version; to appear in Revista Mathematica Iberoamericana, 28

Variations on a theorem of Beurling

We consider functions satisfying the subcritical Beurling's condition, viz., $$\int_{\R^n}\int_{\R^n} |f(x)| |\hat{f}(y)| e^{a |x \cdot y|} \, dx \, dy < \infty$$ for some $0 < a < 1.$ We show that such functions are entire vectors for the Schr\"{o}dinger representations of the Heisenberg group. If an eigenfunction $f$ of the Fourier transform satisfies the above condition we show that the Hermite coefficients of $f$ have certain exponential decay which depends on $a$.Comment: 21 page

On the Hermite expansions of functions from Hardy class

Considering functions $f$ on $\R^n$ for which both $f$ and $\hat{f}$ are bounded by the Gaussian $e^{-{1/2}a|x|^2}, 0 < a < 1$ we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for $O(n)-$finite functions thus extending the one dimensional result of Vemuri.Comment: 22 page