scholarly journals Plancherel's theorem and integration without the lattice condition

1984 ◽  
Vol 42 (1) ◽  
pp. 67-73 ◽  
Author(s):  
Michael Leinert
Keyword(s):  
Plancherel’S Theorem ◽  
Lattice Condition

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 Daniell-Stone integration without the lattice condition

1982 ◽  
Vol 38 (1) ◽  
pp. 258-265 ◽  
Author(s):  
Michael Leinert
Keyword(s):  
Lattice Condition

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.

Finite Rank Operators in Certain Algebras

1999 ◽  
Vol 42 (4) ◽  
pp. 452-462 ◽  
Author(s):  
Sean Bradley
Keyword(s):  
Finite Rank ◽  
Linear Operators ◽  
Bounded Linear ◽  
Finite Rank Operators ◽  
Riesz Operators ◽  
Finite Lattices ◽  
Rank One ◽  
Necessary And Sufficient ◽  
Rank One Operator ◽  
Lattice Condition

AbstractLet Alg(ℒ) be the algebra of all bounded linear operators on a normed linear space X leaving invariant each member of the complete lattice of closed subspaces L. We discuss when the subalgebra of finite rank operators in Alg(ℒ) is non-zero, and give an example which shows this subalgebra may be zero even for finite lattices. We then give a necessary and sufficient lattice condition for decomposing a finite rank operator F into a sum of a rank one operator and an operator whose range is smaller than that of F, each of which lies in Alg(ℒ). This unifies results of Erdos, Longstaff, Lambrou, and Spanoudakis. Finally, we use the existence of finite rank operators in certain algebras to characterize the spectra of Riesz operators (generalizing results of Ringrose and Clauss) and compute the Jacobson radical for closed algebras of Riesz operators and Alg(ℒ) for various types of lattices.

scholarly journals On systems of parabolic variational inequalities with multivalued terms

2020 ◽  
Author(s):  
Siegfried Carl ◽  
Vy. K. Le
Keyword(s):  
Variational Inequalities ◽  
Analytical Framework ◽  
Multivalued Function ◽  
Time Dependent ◽  
Existence Theory ◽  
Hemivariational Inequalities ◽  
Cylindrical Domain ◽  
Upper Semicontinuous ◽  
Quasilinear Elliptic ◽  
Lattice Condition

AbstractIn this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain $$Q=\varOmega \times (0,\tau )$$ Q = Ω × ( 0 , τ ) : For $$k=1,\dots , m$$ k = 1 , ⋯ , m , find $$u_k\in K_k$$ u k ∈ K k and $$\eta _k\in L^{p'_k}(Q)$$ η k ∈ L p k ′ ( Q ) such that $$\begin{aligned}&u_k(\cdot ,0)=0\ \text{ in } \varOmega ,\ \ \eta _k(x,t)\in f_k(x,t,u_1(x,t), \dots , u_m(x,t)), \\&\langle u_{kt}+A_k u_k, v_k-u_k\rangle +\int _Q \eta _k\, (v_k-u_k)\,dxdt\ge 0,\ \ \forall \ v_k\in K_k, \end{aligned}$$ u k ( · , 0 ) = 0 in Ω , η k ( x , t ) ∈ f k ( x , t , u 1 ( x , t ) , ⋯ , u m ( x , t ) ) , ⟨ u kt + A k u k , v k - u k ⟩ + ∫ Q η k ( v k - u k ) d x d t ≥ 0 , ∀ v k ∈ K k , where $$K_k $$ K k is a closed and convex subset of $$L^{p_k}(0,\tau ;W_0^{1,p_k}(\varOmega ))$$ L p k ( 0 , τ ; W 0 1 , p k ( Ω ) ) , $$A_k$$ A k is a time-dependent quasilinear elliptic operator, and $$f_k:Q\times \mathbb {R}^m\rightarrow 2^{\mathbb {R}}$$ f k : Q × R m → 2 R is an upper semicontinuous multivalued function with respect to $$s\in {\mathbb R}^m$$ s ∈ R m . We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints $$K_k$$ K k , systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.

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.

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