Order Structures in Banach Spaces of Analytic Functions

In this paper we consider some Banach spaces of analytic functions on the unit disk generated by the cone of analytic functions with monotone decreasing Taylor coefficients. We get that some of these spaces are Banach lattices with respect to this cone. Different ordered spaces of linear bounded operators acting between the previous spaces are also investigated, with emphasis on the so-called regular multipliers and Hankel operators.     

Toeplitz operators on Dirichlet spaces

In this paper we consider Toeplitz operators on Dirichlet spaces of the unit disk in whose symbols are nonnegative measures. We obtain necessry and sufficient conditions on the symbols for the operator to be bounded and compact. If the symbols are supported in a cone we also get necessary and sufficient conditions for the operators to belong to the Schatten p-class. Application to the Hankel operators are indicated.This work supported in part by NSF grant DMS 8701271     

A look-ahead algorithm for the solution of general Hankel systems

Summary The solution of systems of linear equations with Hankel coefficient matrices can be computed with onlyO(n ²) arithmetic operations, as compared toO(n ³) operations for the general cases. However, the classical Hankel solvers require the nonsingularity of all leading principal submatrices of the Hankel matrix. The known extensions of these algorithms to general Hankel systems can handle only exactly singular submatrices, but not ill-conditioned ones, and hence they are numerically unstable. In this paper, a stable procedure for solving general nonsingular Hankel systems is presented, using a look-ahead technique to skip over singular or ill-conditioned submatrices. The proposed approach is based on a look-ahead variant of the nonsymmetric Lanczos process that was recently developed by Freund, Gutknecht, and Nachtigal. We first derive a somewhat more general formulation of this look-ahead Lanczos algorithm in terms of formally orthogonal polynomials, which then yields the look-ahead Hankel solver as a special case. We prove some general properties of the resulting look-ahead algorithm for formally orthogonal polynomials. These results are then utilized in the implementation of the Hankel solver. We report some numerical experiments for Hankel systems with ill-conditioned submatrices.The research of the first author was supported by DARPA via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).The research of the second author was supported in part by NSF grant DRC-8412314 and Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).     

A new and self‐contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations

We present a new and self‐contained theory for mapping properties of the boundary operators for slit diffraction occurring in Sommerfeld's diffraction theory, covering two different cases of the polarisation of the light. This theory is entirely developed in the context of the boundary operators with a Hankel kernel and not based on the corresponding mixed boundary value problem for the Helmholtz equation. For a logarithmic approximation of the Hankel kernel we also study the corresponding mapping properties and derive explicit solutions together with certain regularity results.     

Hankel operators on the Dirichlet space

We study (small) Hankel operators on the Dirichlet space D with symbols in a class of function space, and show that such (small) Hankel operators are closely related to the corresponding Hankel operators on the Bergman space and the Hardy space H².     

Riordan arrays and the LDU decomposition of symmetric Toeplitz plus Hankel matrices

We examine a result of Basor and Ehrhardt concerning Hankel and Toeplitz plus Hankel matrices, within the context of the Riordan group of lower-triangular matrices. This allows us to determine the LDU decomposition of certain symmetric Toeplitz plus Hankel matrices. We also determine the generating functions and Hankel transforms of associated sequences.     

On the reduction of the normal Hankel problem to two particular cases

The normal Hankel problem (NHP) is to describe complex matrices that are normal and Hankel at the same time. The available results related to the NHP can be combined into two groups. On the one hand, there are several known classes of normal Hankel matrices. On the other hand, the matrix classes that may contain normal Hankel matrices not belonging to the known classes were shown to admit a parametrization by real 2 × 2 matrices with determinant 1. We solve the NHP for the cases where the characteristic matrix W of the given class has: (a) complex conjugate eigenvalues; (b) distinct real eigenvalues. To obtain a complete solution of the NHP, it remains to analyze two situations: (1) W is the Jordan block of order two for the eigenvalue 1; (2) W is the Jordan block of order two for ?1.     

Hankel Operators on Bergman Spaces of Tube Domains over Symmetric Cones

We present here some criteria for Schatten-Von Neumann class membership for the small Hankel operator on Bergman space A ²(T _Ω), when T _Ω is the tube over the symmetric cone Ω. The author would like to thank professor Aline Bonami for helpful advices.     

Essential Norm of Toeplitz Operators and Hankel Operators on the Weighted Bergman Space

In this paper, we show that on the weighted Bergman space of the unit disk the essential norm of a noncompact Hankel operator equals its distance to the set of compact Hankel operators and is realized by infinitely many compact Hankel operators, which is analogous to the theorem of Axler, Berg, Jewell and Shields on the Hardy space in Axler et al. (Ann Math 109:601–612, 1979); moreover, the distance is realized by infinitely many compact Hankel operators with symbols continuous on the closure of the unit disk and vanishing on the unit circle.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.     

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.     

A complete solution of the normal Hankel problem

The normal Hankel problem is the one of characterizing the matrices that are normal and Hankel at the same time. We give a complete solution of this problem.     

On algebras of Toeplitz fuzzy matrices

In this paper, necessary and sufficient conditions are given for a product of Toeplitz fuzzy matrices to be Toeplitz. As an application, a criterion for normality of Toeplitz fuzzy matrices is derived and conditions are deduced for symmetric idempotency of Toeplitz fuzzy matrices. We discuss similar results for Hankel fuzzy matrices. Keywords: Fuzzy matrix, Toeplitz and Hankel matrices.     

Compact Hankel Forms on Planar Domains

A Hankel form on a Hilbert function space is a bounded, symmetric, bilinear form [., .] satisfying [fx, y] = [x, fy] for a class of multipliers f. We prove analogs of Weyl–Horn and Ky Fan inequalities for compact Hankel forms, and apply them to estimate the related eigenvalues, both for Hardy–Smirnov and Bergman spaces norms associated to multiply connected planar domains. In the case of the unit disk, we investigate the asymptotic of some measures constructed by eigenfunctions of Hankel operators with certain Markov functions as symbols. Submitted: May 2, 2008. Accepted: June 28, 2008.     

Continuity of the fractional Hankel wavelet transform on the spaces of type S

In this present article, we study the fractional Hankel transform and its inverse on certain Gel'fand‐Shilov spaces of type S. The continuous fractional wavelet transform is defined involving the fractional Hankel transform. The continuity of fractional Hankel wavelet transform is discussed on Gel'fand‐Shilov spaces of type S. This article goes further to discuss the continuity property of fractional Hankel transform and fractional Hankel wavelet transform on the ultradifferentiable function spaces.     

Hankel matrices for the period-doubling sequence

We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders.     

Weyl transforms associated with the Hankel transform in Clifford analysis

In this paper, we define the Hankel–Wigner transform in Clifford analysis and therefore define the corresponding Weyl transform. We present some properties of this kind of Hankel–Wigner transform, and then give the criteria of the boundedness of the Weyl transform and compactness on the L^p space. Copyright © 2005 John Wiley & Sons, Ltd.     

L p -L q Estimates for Bergman Projections in Bounded Symmetric Domains of Tube Type

Let D be an irreducible bounded symmetric domain of tube type in ? ⁿ . The class of Bloch functions is well known in this context, in connection with Hankel operators or duality of Bergman spaces. Contrary to what happens in the unit ball, Bloch functions do not belong to all Lebesgue spaces L ^p (D) for p<∞ in higher rank. We give here both necessary and sufficient conditions on p for such an embedding. This question is equivalent to local boundedness properties of the Bergman projection in the tube domain over a symmetric cone that is conformally equivalent to D. We are linked to consider L ^∞–L ^q inequalities on symmetric cones, which may be of independent interest, and study more systematically estimates with loss for the Bergman projection. The proofs are based on a very precise estimate on an integral related to the Gamma function of a symmetric cone.     

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.