Real Paley-Wiener Theorems

The aim of this paper is to exhibit a real Paley–Wienerspace sitting inside the Schwartz space, and to give a quickand simple proof of a Paley–Wiener-type theorem. A simpleand elementary proof of a theorem postulated by H. H. Bang isalso given. 2000 Mathematics Subject Classification 42A38.     

Paley-Wiener and Boas Theorems for the Quaternion Fourier Transform

This paper establishes a real Paley-Wiener theorem to characterize the quaternion-valued functions whose quaternion Fourier transform has compact support by the partial derivative and also a Boas theorem to describe the quaternion Fourier transform of these functions that vanish on a neighborhood of the origin by an integral operator.     

Toeplitz and Hankel operators on the Paley-Wiener space

The basic theory of Toeplitz and Hankel operators acting on the Paley-Weiner space is developed. This includes criteria for boundedness, compactness, being of finite rank, and membership in the Schatten-von Neumann ideals. Similar questions are considered for the related operators formed by commuting the discrete Hilbert transform with a multiplication operator.Supported in part by a grant from the National Science Foundation.     

Hankel operators on the Paley-Wiener space in ℝ d

The basic theory of Besov spaces inI ^d of Paley-Wiener type is developed. This kind of Besov spaces turns out to be quite a success to characterize the Schatten-von Neumann ideal criteria for Hankel operators acting on Paley-Wiener spaces inI ^d.     

Real Paley-Wiener theorems for the Clifford Fourier transform

Associated with the Dirac operator and partial derivatives, this paper establishes some real Paley-Wiener type theorems to characterize the Clifford-valued functions whose Clifford Fourier transform (CFT) has compact support. Based on the Riemann-Lebesgue theorem for the CFT, the Boas theorem is provided to describe the CFT of Clifford-valued functions that vanish on a neighborhood of the origin.     

Multiplier Theorems for the Short-Time Fourier Transform

So-called short-time Fourier transform multipliers (also called Anti-Wick operators in the literature) arise by applying a pointwise multiplication operator to the STFT before applying the inverse STFT. Boundedness results are investigated for such operators on modulation spaces and on L _p -spaces. Because the proofs apply naturally to Wiener amalgam spaces the results are formulated in this context. Furthermore, a version of the Hardy-Littlewood inequality for the STFT is derived. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No K67642.     

Real Paley-Wiener theorems associated with the weinstein operator

In this paper we use real analysis techniques to establish a new real Paley-Wiener theorems for the Fourier-Bessel transform associated with the Weinstein operator. More precisely we characterize the C ^∞-functions whose image under the Fourier-Bessel transform are functions with compact support through an L ^p growth condition, p ∈ [1, +∞] and we give another version of the real Paley-Wiener theorem for L ²-functions.     

On Polyconvolutions Generated by the Hankel Transform

In this paper, we construct two polyconvolutions (generalized convolutions) with weight $\gamma = x^{ - \nu }$ generated by the Hankel transform possessing the factorization relations ${\text{H}}_\nu [h_1 ](x) = x^{ - \nu } {\text{H}}_\mu {\text{[}}f](x){\text{H}}_\mu {\text{[}}g](x),{\text{ H}}_\mu [h_1 ](x) = x^{ - \nu } {\text{H}}_\nu {\text{[}}f](x){\text{H}}_\mu {\text{[}}g](x).$ Here H_μ is the Hankel transform operator of order μ. Conditions for the existence of the constructed polyconvolutions are found. On their basis, using the differential properties of the Hankel transform, we obtain two more polyconvolutions. The derived constructions allow us to solve new classes of integral and integro-differential equations and systems of equations.     

Spacetime Causality in the Study of the Hankel Transform

We study Hilbert space aspects of the Klein-Gordon equation in two-dimensional spacetime. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. We apply this to give a novel proof, solely based on the causality of this spatio-temporal wave propagation, of the theorem of de Branges and V. Rovnyak concerning Hankel pairs with a support property. We recover their isometric expansion as an application of Riemann’s general method for solving Cauchy-Goursat problems of hyperbolic type. Communicated by Vincent Rivasseau Submitted: October 28, 2005; Accepted: February 17, 2006     

Ratio Mercerian Theorems with Applications to Hankel and Fourier Transforms

We complete and complement our recent work on Drasin-Shea-Jordantheorems for Fourier and Hankel transforms. In improving onthe methods of our previous work, we were led to certain ratioMercerian theorems for general kernels; these yield definitiveversions of our earlier results for Fourier and Hankel transforms.1991 Mathematics Subject Classification: primary 44A15; secondary47B38.     

On Hankel Transform and Hankel Convolution of Beurling Type Distributions Having Upper Bounded Support

In this paper we study Beurling type distributions in the Hankel setting. We consider the space of Beurling type distributions on (0, ) having upper bounded support. The Hankel transform and the Hankel convolution are studied on the space . We also establish Paley Wiener type theorems for Hankel transformations of distributions in .     

Theorems of the Alternative for Inequality Systems of Real Polynomials

In this paper, we establish theorems of the alternative for inequality systems of real polynomials. For the real quadratic inequality system, we present two new results on the matrix decomposition, by which we establish two theorems of the alternative for the inequality system of three quadratic polynomials under an assumption that at least one of the involved forms be negative semidefinite. We also extend a theorem of the alternative to the case with a regular cone. For the inequality system of higher degree real polynomials, defined by even order tensors, a theorem of the alternative for the inequality system of two higher degree polynomials is established under suitable assumptions. As a byproduct, we give an equivalence result between two statements involving two higher degree polynomials. Based on this result, we investigate the optimality condition of a class of polynomial optimization problems under suitable assumptions.     

The Distributional Hankel Transform of Marcel Riesz's Ultrahyperbolic Kernel

In this article we give a sense to the distributional Hankel transform of Marcel Riesz's ultrahyperbolic kernel. First we evaluate Rα(u) in α = ?2k and α = 2k for the cases μ even and ν odd, μ even and ν even, and μ odd and ν odd, μ odd and ν even, where and Finally in Section 4 we obtain the distributional Hankel transform of Marcel Riesz's ultrahyperbolic kernel.     

Fundamental Theorems of Calculus for Packing Measures on the Real Line

The fundamental Theorems of Calculus are extended to the treatment of packing measures on the real line. These are related to the corresponding result for Hausdorff measures. We prove that the centered inner-envelope derivative and the outer-envelope derivative of a continuous increasing function on an interval differ, but in a certain sense they are almost everywhere linear and the theorems are true for h-continuous almost h-singular functions of bounded variation.     

Weighted Restriction Theorems for the Fourier Transform on a Kind of Surfaces

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实数系基本定理等价性的完全互证

综合给出了实数系六个基本定理的等价性的完全互证方法,并归纳了各种证明方法的规律,旨在把抽象的证明转化为容易掌握的基本方法.     

Poisson Summation Formulas and Inversion Theorems for an Infinite Continuous Legendre Transform

The classical Fourier transform and Fourier series are linked by the Poisson summation formula. The goal of this article is to find an infinite continuous Legendre transform which complements Legendre series in a similar way. To this end, the finite continuous Legendre transform due to Butzer/Stens/Wehrens is extended to an infinite transform. We show that for the new Legendre transform variants of Poissons formula and inversion theorems hold.