Plancherel's Theorem and the Shannon Series Derived Simultaneously

1980 ◽  
Vol 87 (3) ◽  
pp. 193 ◽  
Author(s):  
Antonio Steiner
Keyword(s):  
Plancherel’S Theorem

scholarly journals A Sequential Approach to Mild Distributions

Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 25 ◽  
Author(s):  
Hans G. Feichtinger
Keyword(s):  
Fourier Transform ◽  
Number System ◽  
Continuous Functions ◽  
Sequential Approach ◽  
Tempered Distributions ◽  
Segal Algebra ◽  
Gabor Analysis ◽  
Plancherel’S Theorem ◽  
Lca Groups ◽  
Gelfand Triple

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.

scholarly journals Stone's theorem and completeness of orthogonal systems

1971 ◽  
Vol 12 (2) ◽  
pp. 211-223 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Essential Feature ◽  
Elliptic Functions ◽  
Completeness Theorem ◽  
Polynomial Systems ◽  
Orthogonal Functions ◽  
Jacobian Elliptic Functions ◽  
Special Cases ◽  
Weierstrass Approximation Theorem ◽  
Plancherel’S Theorem ◽  
Stone’S Theorem

It is well known (e.g. Stone [1]) that the Stone-Weierstrass approximation theorem can be used to prove the completeness of various systems of orthogonal polynomials, e.g. Chebyshev polynomials. In this paper, Stone's theorem is used to prove a more general completeness theorem, which includes as special cases Plancherel's theorem, the corresponding theorem for Hankel transforms, the completeness of various polynomial systems, and certain expansions in Jacobian elliptic functions. The essential feature common to all these systems is a certain algebraic structure — if S is an appropriate vector space spanned by orthogonal functions, then the algebra A generated by S is contained in the closure of S in a suitable norm.

scholarly journals Tomography bounds for the Fourier extension operator and applications

2021 ◽  
Author(s):  
Jonathan Bennett ◽  
Shohei Nakamura
Keyword(s):  
Large Mass ◽  
Extension Operator ◽  
Weighted Norm ◽  
Radon Transforms ◽  
X Ray ◽  
Fourier Restriction ◽  
Affine Subspaces ◽  
Natural Variants ◽  
The Fourier Transform ◽  
Plancherel’S Theorem

AbstractWe explore the extent to which the Fourier transform of an $$L^p$$ L p density supported on the sphere in $$\mathbb {R}^n$$ R n can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing bounds on quantities of the form $$X(|\widehat{gd\sigma }|^2)$$ X ( | g d σ ^ | 2 ) and $$\mathcal {R}(|\widehat{gd\sigma }|^2)$$ R ( | g d σ ^ | 2 ) , where X and $$\mathcal {R}$$ R denote the X-ray and Radon transforms respectively; here $$d\sigma $$ d σ denotes Lebesgue measure on the unit sphere $$\mathbb {S}^{n-1}$$ S n - 1 , and $$g\in L^p(\mathbb {S}^{n-1})$$ g ∈ L p ( S n - 1 ) . We also identify some conjectural bounds of this type that sit between the classical Fourier restriction and Kakeya conjectures. Finally we provide some applications of such tomography bounds to the theory of weighted norm inequalities for $$\widehat{gd\sigma }$$ g d σ ^ , establishing some natural variants of conjectures of Stein and Mizohata–Takeuchi from the 1970s. Our approach, which has its origins in work of Planchon and Vega, exploits cancellation via Plancherel’s theorem on affine subspaces, avoiding the conventional use of wave-packet and stationary-phase methods.

scholarly journals Integrability of Fourier transforms under an ergodic hypothesis

1978 ◽  
Vol 26 (2) ◽  
pp. 129-153
Author(s):  
Gavin Brown
Keyword(s):  
Fourier Transforms ◽  
General Setting ◽  
Locally Compact ◽  
Ergodic Hypothesis ◽  
Locally Compact Abelian Groups ◽  
Discrete Random Variables ◽  
Ergodic Condition ◽  
Compact Abelian Groups ◽  
Plancherel’S Theorem ◽  
Underlying Theme

AbstractThe object is to unify and complement some recent theorems of Hewitt and Ritter on the integrability of Fourier transforms, but the underlying theme is the ancient one that Plancherel's theorem is the “only” integrability constraint on Fourier transforms. The distinguishing feature of the results is that we restrict attention to positive measures (or functions) which satisfy an ergodic condition and whose transforms are positive. (In fact we employ sums of discrete random variables, a technique which seems to have been largely ignored in context.) The general setting is that of locally compact abelian groups but we are chiefly interested in the line or the circle, and it appears that the theorems are new for these classical groups.

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