scholarly journals Spherical Means on the Heisenberg Group and a Restriction Theorem for the Symplectic Fourier Transform

1991 ◽  
pp. 135-155 ◽  
Author(s):  
Sundaram Thangavelu
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
Spherical Means ◽  
Restriction Theorem

scholarly journals Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1060
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del del Olmo
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Heisenberg Group ◽  
Real Line ◽  
Hermite Functions ◽  
Unitary Representations ◽  
Weyl Groups ◽  
The Real ◽  
Symmetry Properties ◽  
The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

Some remarks on the restriction of the Fourier transform to surfaces

1993 ◽  
Vol 113 (1) ◽  
pp. 153-159 ◽  
Author(s):  
S. W. Drury ◽  
K. Guo
Keyword(s):  
Fourier Transform ◽  
Riesz Potential ◽  
Restriction Theorem ◽  
The Fourier Transform

AbstractFor a class of kernels, we prove the Lp estimate for the exotic Riesz potential, with which a restriction theorem of the Fourier transform to surfaces of half the ambient dimension is proved. A simpler proof of Barcelo's result is given. We also find that it is possible to combine the Hausdorff–Young theorem with the Fefferman–Zygmund method to prove some optimal results on the restriction theorem.

scholarly journals Revisiting the Fourier transform on the Heisenberg group

2014 ◽  
Vol 58 ◽  
pp. 47-63 ◽  
Author(s):  
R. Lakshmi Lavanya ◽  
S. Thangavelu
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
The Fourier Transform

Heat kernels and Green functions of sub-Laplacians on Heisenberg groups with multi-dimensional center

2018 ◽  
Vol 13 (4) ◽  
pp. 38
Author(s):  
Shahla Molahajloo ◽  
M.W. Wong
Keyword(s):  
Fourier Transform ◽  
Green Function ◽  
Heisenberg Group ◽  
Inverse Fourier Transform ◽  
Heat Kernels ◽  
Hermite Functions ◽  
Green Functions ◽  
Heisenberg Groups ◽  
Explicit Formulas ◽  
Special Hermite Functions

We compute the sub-Laplacian on the Heisenberg group with multi-dimensional center. By taking the inverse Fourier transform with respect to the center, we get the parametrized twisted Laplacians. Then by means of the special Hermite functions, we find the eigenfunctions and the eigenvalues of the twisted Laplacians. The explicit formulas for the heat kernels and Green functions of the twisted Laplacians can then be obtained. Then we give an explicit formula for the heat kernal and Green function of the sub-Laplacian on the Heisenberg group with multi-dimensional center.

The λ-cosine transforms with odd kernel and the hemispherical transform

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Boris Rubin
Keyword(s):  
Fourier Transform ◽  
Unit Sphere ◽  
Borel Measure ◽  
Fractional Integrals ◽  
Inversion Method ◽  
Spherical Means ◽  
Finite Borel Measure ◽  
Basic Facts ◽  
Hemispherical Transform ◽  
The Fourier Transform

AbstractWe review some basic facts about the λ-cosine transforms with odd kernel on the unit sphere S n−1 in ℝn. These transforms are represented by the spherical fractional integrals arising as a result of evaluation of the Fourier transform of homogeneous functions. The related topic is the hemispherical transform which assigns to every finite Borel measure on S n−1 its values for all hemispheres. We revisit the known facts about this transform and obtain new results. In particular, we show that the classical Funk- Radon-Helgason inversion method of spherical means is applicable to the hemispherical transform of L p-functions.

scholarly journals A characterisation of the Fourier transform on the‎ ‎Heisenberg group

2012 ◽  
Vol 3 (1) ◽  
pp. 109-120 ◽  
Author(s):  
R‎. ‎Lakshmi Lavanya ◽  
S‎. ‎Thangavelu
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
The Fourier Transform

The Vacuum Distributions of the Truncated Virasoro Fields are Products of Gamma Distributions

2017 ◽  
Vol 24 (01) ◽  
pp. 1750004 ◽  
Author(s):  
Luigi Accardi ◽  
Andreas Boukas ◽  
Yun-Gang Lu
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Heisenberg Group ◽  
Lie Algebra ◽  
Fock Space ◽  
Random Variables ◽  
Virasoro Algebra ◽  
Commutation Relations ◽  
Gamma Distributions

In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.

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