Displaying similar documents to “Spherical means on the Heisenberg group and a restriction theorem for the symplectic Fourier transform.”

Fourier transform of Schwartz functions on the Heisenberg group

Francesca Astengo, Bianca Di Blasio, Fulvio Ricci (2013)

Studia Mathematica

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Let H₁ be the 3-dimensional Heisenberg group. We prove that a modified version of the spherical transform is an isomorphism between the space 𝓢ₘ(H₁) of Schwartz functions of type m and the space 𝓢(Σₘ) consisting of restrictions of Schwartz functions on ℝ² to a subset Σₘ of the Heisenberg fan with |m| of the half-lines removed. This result is then applied to study the case of general Schwartz functions on H₁.

A restriction theorem for the Heisenberg motion

P. Ratnakumar, Rama Rawat, S. Thangavelu (1997)

Studia Mathematica

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We prove a restriction theorem for the class-1 representations of the Heisenberg motion group. This is done using an improvement of the restriction theorem for the special Hermite projection operators proved in [13]. We also prove a restriction theorem for the Heisenberg group.

Asymptotic spherical analysis on the Heisenberg group

Jacques Faraut (2010)

Colloquium Mathematicae

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The asymptotics of spherical functions for large dimensions are related to spherical functions for Olshanski spherical pairs. In this paper we consider inductive limits of Gelfand pairs associated to the Heisenberg group. The group K = U(n) × U(p) acts multiplicity free on 𝓟(V), the space of polynomials on V = M(n,p;ℂ), the space of n × p complex matrices. The group K acts also on the Heisenberg group H = V × ℝ. By a result of Carcano, the pair (G,K) with G = K ⋉ H is a Gelfand pair....

A Paley-Wiener theorem on NA harmonic spaces

Francesca Astengo, Bianca di Blasio (1999)

Colloquium Mathematicae

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Let N be an H-type group and consider its one-dimensional solvable extension NA, equipped with a suitable left-invariant Riemannian metric. We prove a Paley-Wiener theorem for nonradial functions on NA supported in a set whose boundary is a horocycle of the form Na, a ∈ A.

L-bounds for spherical maximal operators on Z.

Akos Magyar (1997)

Revista Matemática Iberoamericana

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We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Z. We decompose the discrete spherical measures as an integral of Gaussian kernels s(x) = e. By using Minkowski's integral inequality it is enough to prove L-bounds for the corresponding convolution operators. The proof is then based on L-estimates by analysing the Fourier transforms ^s(ξ), which can be handled by making use of the circle method for exponential sums. As a corollary one...

An application of shift operators to ordered symmetric spaces

Nils Byrial Andersen, Jérémie M. Unterberger (2002)

Annales de l’institut Fourier

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We study the action of elementary shift operators on spherical functions on ordered symmetric spaces ℳ m , n of Cayley type, where m ∈ ℕ denotes the multiplicity of the short roots and n ∈ ℕ the rank of the symmetric space. For m even we apply this to prove a Paley-Wiener theorem for the spherical Laplace transform defined on ℳ m , n by a reduction to the rank 1 case. Finally we generalize our notions and results to B C n type root systems.