The Evolution of Applied Harmonic Analysis

Prestini, Elena

doi:10.1007/978-0-8176-8140-1

Cited by 14 publications

(13 citation statements)

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“…References for the Radon transform include Bracewell [62], Dym and McKean [147], Gelfand, Graev, and Vilenkin [204], Herman [290], Helgason [279], Louis [431,432], Ludwig [435], Elena Prestini [522], Quinto [524], and Shepp and Kruskal [588]. In fact, Funk [187] proved the analogue of (2.32) for S 2 rather than S 1 one year earlier than Radon proved his result.…”

Section: Cat Scanners and The Radon Transformmentioning

confidence: 99%

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Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Terras

2013

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Whenever there is a large earthquake the Earth vibrates for days afterwards. The vibrations consist of the superposition of the elastic-gravitational normal modes of the Earth that are excited by the earthquake.-From F. Gilbert [212, p. 107].A (surface or Laplace) spherical harmonic is an eigenfunction of the Laplacian on the sphere. These are the analogues of exponentials for Fourier analysis on the sphere. Laplace and Legendre introduced these functions in order to study gravitational theory in the 1780s. Spherical harmonics are necessary for the analysis of any phenomena with spherical symmetry; e.g., earthquakes, the hydrogen atom, and the solar corona. Some of these topics will be discussed later in this section.Since our treatment of harmonic analysis on the sphere is rather condensed, the reader may want to consult some of the following references for more information: Before discussing spherical harmonics, we need to understand something about the geometry of the sphere. This symmetric space is closely related to the orthogonal group O(n) of real n × n matrices U such that t UU = I, where t U denotes the transpose of U and I denotes the n × n identity matrix. The special orthogonal SO(n), is the subgroup of matrices U in O(n) such that the determinant of U is one. You can regard U in SO(n − 1) as an element of SO(n) by formingThe group O(n) is a compact group and the sphere is a compact symmetric space, making some of the analysis on it somewhat easier.Exercise 2.1.1 (The Sphere as a Quotient or Homogeneous Space). Consider the sphere S n−1 = {x ∈ R n | x = 1}. Show that S n−1 can be identified with the quotient space SO(n)/SO(n − 1), using the preceding identification of SO(n − 1) as a subgroup of SO(n). Hint.Map the coset gSO(n − 1) to the vector ge n for g in SO(n − 1) and e n = t (0,... ,0, 1).You may also want to read the discussion of the topology of spheres and orthogonal groups in Chevalley [85,. In particular, the fundamental (or Poincaré) group of SO(2) is isomorphic to Z, while that of SO(n), n ≥ 3, has order 2.From now on we shall consider only the sphere S 2 . See the references for the general case. Now the sphere S 2 is a differentiable manifold. This means that locally it looks like two-dimensional Euclidean space. To make this precise, we use the usual angular coordinates (θ , ϕ), 0 < ϕ < 2π, 0 < θ < π, pictured in Fig. 2.1, to parameterize S 2 except for the semi-circle C through the poles and (1, 0, 0). A similar coordinate patch can be constructed to cover the rest of the sphere. The equations for the rectangular coordinates (x, y, z) of a point on S 2 in terms of the angular coordinates are x = sin θ cos ϕ y = sin θ sin ϕ z = cos θ ⎫ ⎬ ⎭ (2.1)

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Section: Cat Scanners and The Radon Transformmentioning

confidence: 99%

“…In 2003, the Nobel Prize for Physiology or Medicine went to Paul Lauterbur and Sir Peter Mansfields for their work on MRI. See Elena Prestini [522], Chap. 8, for more details on the subject.…”

Section: Exercise 2212mentioning

confidence: 99%

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Terras

2013

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“…(20). The reader should address in particular the book of Prestini [18] where the presented approach is developed in a less general way but with very interesting practical applications. Restricting the attention to the heat conduction problem, many authors deal with the cases of constant and periodic heating [19 -26] though the general problem is not discussed in detail.…”

Section: General Solution By Means Of Fourier Analysismentioning

confidence: 99%

Diffusion through a half space: equivalence between different formulations of the unique solution

Mantegna¹,

Colombo²

2014

ams

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Diffusion through a half space involves a classical parabolic partial differential equation that is encountered in many fields of physics and has significant engineering applications, concerning particularly heat and mass transfer. However, in the specialized literature, the solution is usually achieved restricting the problem to particular cases and attaining apparently different formulations, thus a comprehensive overview is hindered. In this paper, the solution of the diffusion equation in a half space with a boundary condition of the first kind is worked out by means of the Fourier's Transform, the Green's function and the similarity variable, with a proof of equivalencenot found elsewhereof these different approaches. The keystone of the proof rests on the square completion method applied to Gaussian-like integrals, widely used in Quantum Field Theory.

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“…Here x, y is the dot product of the vectors x and y in R d . The many applications of the FT and its discrete variants are well-documented (see for example [14]). Among the many reasons for the wide applicability of the FT, we include its attractive covariance properties, its energy conservation property, and the simple closed form of its kernel.…”

Section: Fractional Fourier Transforms and The Hermite Functions Thementioning

confidence: 99%

The Fractional Clifford-Fourier Kernel

Craddock

Hogan²

2013

J Fourier Anal Appl

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Abstract. The Clifford-Fourier transform was introduced by Brackx, De Schepper and Sommen and its kernel was computed in dimension d = 2 by the same authors. Here we compute the kernel of a fractional version of the transform when d = 2 and 4. In doing so we solve appropriate wave-type problems on spheres in two and four dimensions. We also give formulae for the solutions of these problems in all even dimensions and hence a means of computing the kernels of the fractional Clifford-Fourier kernels in even dimensions.

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The Evolution of Applied Harmonic Analysis

Cited by 14 publications

References 0 publications

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Diffusion through a half space: equivalence between different formulations of the unique solution

The Fractional Clifford-Fourier Kernel

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