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.

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.

scholarly journals Hermite functions and special Hermite functions  depending on Heisenberg group (ℍⁿ ): دوال هرميت ودوال هرميت الخاصة اعتماداً على زمرة هايزنبرغ (ℍⁿ)

2020 ◽  
Vol 4 (2) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Representation Theory ◽  
Harmonic Oscillator ◽  
Heisenberg Group ◽  
Hermite Functions ◽  
Hermite Operator ◽  
Twisted Laplacian ◽  
Special Hermite Functions ◽  
Hermite Operators ◽  
The Relationship ◽  
Hermite Expansions

In this paper, we talk about Heisenberg group, the most known example from the lie groups. After that, we talk about the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operator and momentum operators (ors). relationship between the representation theory of the Heisenberg group and the position and momentum, that shows how we will make the connection between the Heisenberg group and physics. Then we introduce and study some properties of the Hermite and special Hermite functions. These functions are eigenfunctions of the Hermite and special Hermite operators, respectively. The Hermite operator is often called the harmonic oscillator and the special Hermite operator is sometimes called the twisted Laplacian. As we will later see, the two operators are directly related to the sub-laplacian on the Heisenberg group. The theory of Hermite and special Hermite expansions is intimately connected to the harmonic analysis on the Heisenberg group. They play an important role in our understanding of several problems on ℍⁿ .

scholarly journals Construction of Frames on the Heisenberg Groups

2020 ◽  
Vol 8 (1) ◽  
pp. 382-395
Author(s):  
Der-Chen Chang ◽  
Yongsheng Han ◽  
Xinfeng Wu
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
Operator Theory ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Identity Operator ◽  
Heisenberg Groups ◽  
Zygmund Operator ◽  
The Fourier Transform

Abstract In this paper, we present a construction of frames on the Heisenberg group without using the Fourier transform. Our methods are based on the Calderón-Zygmund operator theory and Coifman’s decomposition of the identity operator on the Heisenberg group. These methods are expected to be used in further studies of several complex variables.

scholarly journals Hermite Functions and Fourier Series

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 853
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano del Olmo
Keyword(s):  
Fourier Transform ◽  
Hermite Functions ◽  
Quantum Oscillator ◽  
Linear Combinations ◽  
Ladder Operators ◽  
Orthonormal Bases ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
The One ◽  
Square Integrable Functions

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

On modified Green functions in exterior problems for the Helmholtz equation

1982 ◽  
Vol 383 (1785) ◽  
pp. 313-332 ◽  
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Helmholtz Equation ◽  
Least Squares ◽  
Present Note ◽  
Boundary Integral ◽  
Green Functions ◽  
Exterior Problems ◽  
The Green Function ◽  
Definition Of

The question of non-uniqueness in boundary integral equation formu­lations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of co­efficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

Green function formulation of the Dirac field in curved space

1966 ◽  
Vol 294 (1439) ◽  
pp. 437-448 ◽  
Keyword(s):  
Green Function ◽  
Dirac Field ◽  
Green Functions ◽  
Field Equations ◽  
Curved Space ◽  
Particle Field ◽  
Function Formulation ◽  
Essential Idea ◽  
Mass Field

A Green function formulation of the Dirac field in curved space is considered in the cases where the mass is constant and where it is regarded as a direct particle field in the manner of Hoyle & Narlikar (1964 c ). This description is equivalent to, and in some ways more satisfactory than, that given in terms of a suitable Lagrangian, in which the Dirac or the mass field is regarded as independent of the geometry. The essential idea is to define the Dirac or the mass field in terms of certain Green functions and sources so that the field equations are satisfied identically, and then to obtain the contribution of these fields to the metric field equations from the variation of a suitable action that is defined in terms of the Green functions and sources.

scholarly journals Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
David W. Pravica ◽  
Njinasoa Randriampiry ◽  
Michael J. Spurr
Keyword(s):  
Fourier Transform ◽  
Theta Function ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Inverse Fourier Transform ◽  
Recursion Relations ◽  
Jacobi Theta Function ◽  
Convergence Results ◽  
The Family ◽  
Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation System

2020 ◽  
Author(s):  
Sen Zhang ◽  
Dingxi Wang ◽  
Yi Li ◽  
Hangkong Wu ◽  
Xiuquan Huang
Keyword(s):  
Fourier Transform ◽  
Stability Analysis ◽  
Condition Number ◽  
Inverse Fourier Transform ◽  
Real Life ◽  
Equation System ◽  
Time Instant ◽  
Transform Matrix ◽  
Spectral Equation ◽  
The Impact

Abstract The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

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