Analysis of Spherical Symmetries in Euclidean Spaces

Müller, Claus

doi:10.1007/978-1-4612-0581-4

Cited by 169 publications

(131 citation statements)

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“…The last part of this article demonstrates how one can use the expansion to obtain results on the Fourier transform of functions which are products of radial factor and homogeneous polynomial. The formulae we obtain encompass both the classical Bochner identity [1,9,5], as well as its generalizations obtained by one of the authors (A.S.) in [11] and more recently by F. J. Gonzalez Vieli in [6]. The detailed presentation of the results contained here is given in the forthcoming papers [2,3] of the authors.…”

Section: Introductionmentioning

confidence: 62%

“…It is concerned with determining the most general representation of the Fourier transform for functions possessing the rotational symmetry of a finite-dimensional representation of the rotation group SO(d). This problem is usually presented in a way which relies on the various types of integral identities related to the Hecke-Funk identity as in [5], or the Poisson integral representation of Bessel functions as in [1,9]. In this approach the expansion of the plane wave into spherical harmonics is deduced as a corollary from these integral identities.…”

Section: Introductionmentioning

confidence: 99%

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A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms

Bezubik¹,

Dbrowska²,

Strasburger³

2004

JNMP

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This article summarizes a new, direct approach to the determination of the expansion into spherical harmonics of the exponential e i(x|y) with x, y ∈ R d . It is elementary in the sense that it is based on direct computations with the canonical decomposition of homogeneous polynomials into harmonic components and avoids using any integral identities. The proof makes also use of the standard representation theoretic properties of spherical harmonics and the explicit form of the reproducing kernels for these spaces by means of classical Gegenbauer polynomials. In the last section of the paper a new method of computing the Fourier transforms of SO(d)-finite functions on the unit sphere is presented which enables us to reobtain both the classical Bochner identity and generalizations of it due to one of the present authors and F. J. Gonzalez Vieli.

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Section: Introductionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 99%

A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms

Bezubik¹,

Dbrowska²,

Strasburger³

2004

JNMP

View full text Add to dashboard Cite

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“…The other properties we obtain are related to the Funk-Hecke formula ( [5], [6]) and with properties of bi-orthogonal systems in the polynomial spaces endowed with the inner product in (1.10). All the results mentioned above form the contents of Sections 2 and 3.…”

Section: Introductionmentioning

confidence: 78%

Orthogonal Bases for Spaces of Complex Spherical Harmonics

Menegatto¹,

Oliveira²

2005

Journal of Applied Analysis

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Abstract. This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω2q of C q . The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L 2 (Ω2q). As a bypass, we study the inner product [f, g] = f (D)(g(z))(0) over the space P(C q ) of polynomials in the variables z, z ∈ C q , in which f (D) is the differential operator with symbol f (z). On the spaces of spherical harmonics, it is shown that the inner product [·, ·] reduces to a multiple of the L 2 (Ω2q) inner product. Bi-orthogonality in (P(C q ), [·, ·]) is fully investigated.

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“…Dick Askey has informed us that this formula follows immediately from the Funk-Hecke theorem [1] and a familiar integral giving the Bessel functions as a Fourier transform. Bochner essentially has this in his book [17]; also see Claus Müller's lectures on spherical harmonics [43] and [44]; it is probably in some notes of Calderon published in Argentina, but they are not widely available. Also see [10], [11], and recent papers [15] and [16].…”

Section: Expansion Formula For a Plane Wavementioning

confidence: 99%

Solution of the Cauchy problem for a time-dependent Schrödinger equation

Meiler

Cordero-Soto

Suslov

2008

Journal of Mathematical Physics

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Abstract. We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU (1, 1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner-Pollaczek polynomials, a Hankel-type integral transform and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as a by-product.

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Analysis of Spherical Symmetries in Euclidean Spaces

Cited by 169 publications

References 0 publications

A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms

A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms

Orthogonal Bases for Spaces of Complex Spherical Harmonics

Solution of the Cauchy problem for a time-dependent Schrödinger equation

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