Banach Gelfand Triples for Applications in Physics and Engineering

Feichtinger, Hans G.

doi:10.1063/1.3183542

Cited by 18 publications

(15 citation statements)

References 90 publications

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“…Many other deeper regularity properties are satisfied by Bopp operators [3]; it turns out that Feichtinger's modulation [9,10] spaces are particularly well-adapted for the study of these properties, and allow a precise analysis of the continuous spectrum using Gelfand triples (see the recent article [11] for a discussion of these topics).…”

Section: Two Weyl Operators After a Fewmentioning

confidence: 97%

A pseudodifferential calculus on non-standard symplectic space

Gosson

2011

Applicable Analysis

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We introduce a class of pseudodifferential operators e A ! acting on functions defined on an arbitrary symplectic space (R 2n , !). These operators arise naturally when one considers the generalized commutation relations from non-commutative quantum mechanics. The connection with the usual Weyl operator bA with symbol a is made using a family of intertwiners W g defined in terms of the cross-Wigner transform W( f, g). We show that if a belongs to some adequate Shubin symbol classes there is a simple relation between the eigenvalues of e A and those of b A.

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Section: Two Weyl Operators After a Fewmentioning

confidence: 97%

A pseudodifferential calculus on non-standard symplectic space

Gosson

2011

Applicable Analysis

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“…j , the normalized Gauss function, which belongs to S(R d ) ⊂ S 0 (R d ) and is Fourier invariant, i.e., we have g = g 0 = g 0 = h and our assumptions are satisfied. Putting all these observations together we get: For any sequence (ρ n ) n≥1 with lim n→∞ ρ n = 0 the sequence (27) or alternativelyf…”

Section: The Functional Analytic Viewpointmentioning

confidence: 99%

“…An alternative approach showing how to introduce S 0 (R d ), • S 0 based on Riemann integrals is provided in [26]. The content of [27,28] show how it could be used for the teaching of engineers and physicists.…”

Section: Remarkmentioning

confidence: 99%

A Sequential Approach to Mild Distributions

Feichtinger

2020

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The Banach Gelfand Triple ( S 0 , L 2 , S 0 ′ ) ( R d ) consists of S 0 ( R d ) , ∥ · ∥ S 0 , a very specific Segal algebra as algebra of test functions, the Hilbert space L 2 ( R d ) , ∥ · ∥ 2 and the dual space S 0 ′ ( R d ) , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce S 0 ( R d ) , ∥ · ∥ S 0 and hence ( S 0 ′ ( R d ) , ∥ · ∥ S 0 ′ ) , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra S 0 ( R d ) , ∥ · ∥ S 0 can be used to establish this natural identification.

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“…7. In the last years the RHS have appeared associated to time-frequency analysis and Gabor analysis that have many applications in physics and engineering related to signal proceesing [43][44][45][46][47][48][49][50]. In particular, applications in electrical engineering have been introduced in [52][53][54].…”

Section: Rigged Hilbert Spacesmentioning

confidence: 99%

Groups, Special Functions and Rigged Hilbert Spaces

Celeghini

Gadella

Olmo

2019

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We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space H and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space, instead they are functionals on the dual space, Φ × , of a rigged Hilbert space, Φ ⊂ H ⊂ Φ × . As a matter of fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can be often continuous operators on Φ with its own topology, so that they admit continuous extensions to the dual Φ × and, therefore, act on the elements of the continuous basis. We have investigated this formalism to various examples of interest in quantum mechanics. In particular, we have considered, SO(2) and functions on the unit circle, SU (2) and associated Laguerre functions, Weyl-Heisenberg group and Hermite functions, SO(3, 2) and spherical harmonics, su(1, 1) and Laguerre functions, su(2, 2) and algebraic Jacobi functions and, finally, su(1, 1)⊕su(1, 1) and Zernike functions on a circle.

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Banach Gelfand Triples for Applications in Physics and Engineering

Cited by 18 publications

References 90 publications

A pseudodifferential calculus on non-standard symplectic space

A pseudodifferential calculus on non-standard symplectic space

A Sequential Approach to Mild Distributions

Groups, Special Functions and Rigged Hilbert Spaces

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