On Marcel Riesz's Ultrahyperbolic Kernel

Let t = (t₁, …, t_n) be a point of ?ⁿ. We shall write . We put by definition R_α(u) = u^(α?n)/2/K_n(α); here α is a complex parameter, n the dimension of the space, and K_n(α) is a constant. First we evaluate □R_α(u) = R_α(u), where □ the ultrahyperbolic operator. Then we obtain the following results: R_?2k(u) = □^kδ; R₀(u) = δ; and □^kR_2k(u) = δ, k = 0, 1, …. The first result is the n-dimensional ultrahyperbolic correlative of the well-known one-dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously.     

The Distributional Hankel Transform of δ(k)(m2×P

In this note we give a sense to certain kinds of n-dimensional distributional Hankel transforms of the Dirac measure δ^(k)(m² + P). The most important result is the interchange formula between the product and the convolution of the Hankel transform of δ^(k)(m² + P).     

The Kernel of a Hankel Operator on the Bergman Space

In this paper we characterise the kernel of a little Hankeloperator on the Bergman space in terms of the inner divisors, and obtain a characterisationfor finite rank little Hankel operators using the invariantsubspace theory technique. 1991 Mathematics Subject Classification47B35.     

Real Paley-Wiener Theorems for the Hankel Transform

We first characterize the image of the compactly supported smooth even functions under the Hankel transform as a subspace of the Schwartz space. We then describe the space of smooth L^p-functions whose Hankel transform has compact support as a subspace of the space of L^p-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.     

On Some Integral Equations with a Hankel Function Kernel

A method is presented for converting integral and integro-differentialequations whose kernel is the Hankel function into Cauchy singular equations. The latter may be solved bystandard function-theoretic methods or by implementing a secondtransformation to convert them into Abel-type equations, whichare readily inverted. Examples are drawn from wave diffractiontheory in two dimensions and include a mixed boundary-valueproblem which may be reduced to a pair of coupled integro-differentialequations. It is shown that the conversion to a singular equationpair and thence to Abel form permits uncoupling and a directsolution. Other applications of the method are briefly indicated.     

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 .     

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     

再生核与小波变换

在再生核基本理论的基础上,介绍了再生核在小波变换中的作用,并且根据连续小波变换像空间是再生核Hilbert空间这一基本事实,借助再生核理论的特殊技巧,建立了Littlewood-Paley和Haar小波变换像空间的再生核函数与已知再生核空间的再生核的关系,为小波变换像空间的进一步研究提供理论基础.     

Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on ? ⁿ . The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander’s L ² estimates for the $\bar{\partial}$ operator are key ingredients in the proof.     

Stability of Inverse Problems for Ultrahyperbolic Equations

In this paper, the authors consider inverse problems of determining a coefficient or a source term in an ultrahyperbolic equation by some lateral boundary data. The authors prove Hlder estimates which are global and local and the key tool is Carleman estimate.     

Iso-Huygens Deformations of the Ultrahyperbolic Operator

New examples of iso-Huygens deformations of the ultrahyperbolic operator and its powers with Calogero–Moser and Lagnese–Stellmacher potentials are considered. Bibliography: 12 titles.     

The Hankel pencil conjecture

The Toeplitz pencil conjecture stated in [W. Schmale, P.K. Sharma, Problem 30-3: singularity of a toeplitz matrix, IMAGE 30 (2003); W. Schmale, P.K. Sharma, Cyclizable matrix pairs over and a conjecture on toeplitz pencils, Linear Algebra Appl. 389 (2004) 33-42] is equivalent to a conjecture for n×n Hankel pencils of the form H_n(x)=(c_i+j-n+1), where c₀=x is an indeterminate, c_l=0 for l<0, and for l1. In this paper it is shown to be implied by another conjecture, which we call the root conjecture. The root conjecture asserts a strong relationship between the roots of certain submaximal minors of H_n(x) specialized to have c₁=c₂=1. We give explicit formulae in the c_i for these minors and prove the root conjecture for minors m_nn,m_n-1,n of degree 6. This implies the Hankel Pencil conjecture for matrices up to size 8×8. The main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous, and use of the Sylvester identity for matrices.     

The Finite Hankel Transform Operator: Some Explicit and Local Estimates of the Eigenfunctions and Eigenvalues Decay Rates

For fixed real numbers \(c>0,\)\(\alpha >-\frac{1}{2},\) the finite Hankel transform operator, denoted by \(\mathcal {H}_c^{\alpha }\) is given by the integral operator defined on \(L^2(0,1)\) with kernel \(K_{\alpha }(x,y)= \sqrt{c xy} J_{\alpha }(cxy).\) To the operator \(\mathcal {H}_c^{\alpha },\) we associate a positive, self-adjoint compact integral operator \(\mathcal Q_c^{\alpha }=c\, \mathcal {H}_c^{\alpha }\, \mathcal {H}_c^{\alpha }.\) Note that the integral operators \(\mathcal {H}_c^{\alpha }\) and \(\mathcal Q_c^{\alpha }\) commute with a Sturm-Liouville differential operator \(\mathcal D_c^{\alpha }.\) In this paper, we first give some useful estimates and bounds of the eigenfunctions \(\varphi ^{(\alpha )}_{n,c}\) of \(\mathcal H_c^{\alpha }\) or \(\mathcal Q_c^{\alpha }.\) These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator \(\mathcal D_c^{\alpha }.\) If \((\mu _{n,\alpha }(c))_n\) and \(\lambda _{n,\alpha }(c)=c\, |\mu _{n,\alpha }(c)|^2\) denote the infinite and countable sequence of the eigenvalues of the operators \(\mathcal {H}_c^{(\alpha )}\) and \(\mathcal Q_c^{\alpha },\) arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer \(n\ge 0,\)\(\lambda _{n,\alpha }(c)\) is decreasing with respect to the parameter \(\alpha .\) As a consequence, we show that for \(\alpha \ge \frac{1}{2},\) the \(\lambda _{n,\alpha }(c)\) and the \(\mu _{n,\alpha }(c)\) have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work.     

Distributional Fourier Transform and Convolution Associated to Chébli-Trimèche Hypergroups

 In this paper we investigate the convolution and the generalized Fourier transform related to Chébli-Trimèche hypergroups on new spaces of distributions. Boundedness, smoothness, uniqueness, and inversion theorems are established for this transform, as well as the main properties of the convolution. The theory developed is used in solving a differential equation involving a singular differential operator.     

Distributional Fourier Transform and Convolution Associated to Chébli-Trimèche Hypergroups

 In this paper we investigate the convolution and the generalized Fourier transform related to Chébli-Trimèche hypergroups on new spaces of distributions. Boundedness, smoothness, uniqueness, and inversion theorems are established for this transform, as well as the main properties of the convolution. The theory developed is used in solving a differential equation involving a singular differential operator. (Received 26 January 2000; in final form 24 July 2001)     

The Smallest Eigenvalue of Hankel Matrices

Let ℋ _N =(s _n+m ),0≤n,m≤N, denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behavior of the smallest eigenvalue λ _N of ℋ _N . It is proven that λ _N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of λ _N can be arbitrarily slow or arbitrarily fast in a sense made precise below. In the indeterminate case, where λ _N is known to be bounded below by a strictly positive constant, we prove that the limit of the nth smallest eigenvalue of ℋ _N for N→∞ tends rapidly to infinity with n. The special case of the Stieltjes–Wigert polynomials is discussed.     

The conditions that the product of Hankel operators is also a Hankel operator

Archiv der Mathematik - In [2], A. Brown and P. R. Halmos gave the necessary and sufficient condition that the product of two Toeplitz operators is also a Toeplitz operator. In this paper, we shall...