Constructive methods for certain Bergman operators

This paper is concerned with the class of linear partial differential equations of second order such that there exist Bergman operators with polynomial kernels (cf, [12]). In an earlier paper [ll] the authors have shown that these equations also admit differential operators as introduced by K. W. Bauer [I]. In the present paper, relations between different types of representations of solutions are investigated. These representations are of interest in developing a function theory of solutions; cf., for instance, K. W. Bauer [I] and S. Ruscheweyh [19]. They are also essential to global extensions of local results obtained by means of Bergman operators of the first kind. The inversion problem for those operators is solved, and it is shown that all solutions of equations of that class which are holomorphic in a domain of C² can be represented by operators with polynomial kernels. Furthermore, a construction principle for deriving the equations investigated by K. W. Bauer [2] is obtained; this yields corresponding representations of solutions by differential and integral operators in a systematic fashion     

Coherent Transforms and Irreducible Representations Corresponding to Complex Structures on a Cylinder and on a Torus

We study a class of algebras with non-Lie commutation relations whose symplectic leaves are surfaces of revolution: a cylinder or a torus. Over each of such surfaces we introduce a family of complex structures and Hilbert spaces of antiholomorphic sections in which the irreducible Hermitian representations of the original algebra are realized. The reproducing kernels of these spaces are expressed in terms of the Riemann theta function and its modifications. They generate quantum Kähler structures on the surface and the corresponding quantum reproducing measures. We construct coherent transforms intertwining abstract representations of an algebra with irreducible representations, and these transforms are also expressed via the theta function.     

Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one

We show some integral representations of the heat kernels and explicit expressions of the Green functions for the Laplace–Beltrami operators on three series of hyperbolic spaces.     

Kernel based approximation in Sobolev spaces with radial basis functions

In this paper, we study several radial basis function approximation schemes in Sobolev spaces. We obtain an optional error estimate by using a class of smoothing operators. We also discussed sufficient conditions for the smoothing operators to attain the desired approximation order. We then construct the smoothing operators by some compactly supported radial kernels, and use them to approximate Sobolev space functions with optimal convergence order. These kernels can be simply constructed and readily applied to practical problems. The results show that the approximation power depends on the precision of the sampling instrument and the density of the available data.     

Branching problems for semisimple Lie groups and reproducing kernels

For a semisimple Lie group G satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for these when restricted to a subgroup H of the same type by combining the classical results with the recent work of T. Kobayashi. We analyze aspects of having differential operators being symmetry-breaking operators; in particular, we prove in the so-called admissible case that every symmetry breaking (H-map) operator is a differential operator. We prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernels. Our techniques are based on realizing discrete series representations as kernels of elliptic invariant differential operators.     

Differential operators for a scale of Poisson type kernels in the unit disc

In this paper, we introduce a scale of differential operators which is shown to correspond canonically to a certain scale of solution kernels generalizing the classical Poisson kernel for the unit disc. The scale of kernels studied is very natural and appears in many places in mathematical analysis, such as in the theory of integral representations of biharmonic functions in the unit disc.     

Locally stationary stochastic processes and Weyl symbols of positive operators

The paper treats locally stationary stochastic processes. A connection with the Weyl symbols of positive operators is observed and explored. We derive necessary conditions on the two functions that constitute the covariance function of a locally stationary stochastic process, some of which use this connection to time-frequency analysis and pseudodifferential operators. Finally, we discuss briefly the subclass of Cohen’s class of time–frequency representations having separable kernels, which is related to locally stationary stochastic processes.     

On periodic matrix-valued Weyl-Titchmarsh functions

We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl-Titchmarsh matrix-valued functions as well as a new version of the functional model for such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions, we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extension which generate a periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach, we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.     

The Reproducing Kernel Structure Arising from a Combination of Continuous and Discrete Orthogonal Polynomials into Fourier Systems

We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier-type systems.We prove Ismail’s conjecture regarding the existence of a reproducing kernel structure behind these kernels, by establishing a link with Saitoh’s theory of linear transformations in Hilbert space. The results are illustrated with Fourier kernels with ultraspherical, their continuous q-extensions and generalizations. As a byproduct of this approach, a new class of sampling theorems is obtained, as well as Neumann-type expansions in Bessel and q-Bessel functions.     

Generalized power functions

In this paper some relations for the kernels of the Carleman–Vekua equation, in particular the representations of these kernels in the form of generalized power functions completely analogous to the well-known elementary Cauchy kernel expansion, are studied. The obtained results are applied to some problems of the theory of generalized analytic functions.     

Properties of representations of operators acting between spaces of vector-valued functions

A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued L ¹-spaces into L ^∞-spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on L ^p -spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations.     

A Domain Integral Equation for the Bergman Kernel

Certain integral operators involving the Szegö, the Bergman and the Cauchy kernels are known to have the reproducing property. Both the Szegö and the Bergman kernels have series representations in terms of an orthonormal basis. In this paper we derive the Cauchy kernel by means of biorthogonality. The ideas involved are then applied to construct a non-Hermitian kernel admitting a reproducing property for a space associated with the Bergman kernel. The construction leads to a domain integral equation for the Bergman kernel.^{1 2}     

The Higher Spin Laplace Operator

This paper deals with a certain class of second-order conformally invariant operators acting on functions taking values in particular (finite-dimensional) irreducible representations of the orthogonal group. These operators can be seen as a generalisation of the Laplace operator to higher spin as well as a second-order analogue of the Rarita-Schwinger operator. To construct these operators, we will use the framework of Clifford analysis, a multivariate function theory in which arbitrary irreducible representations for the orthogonal group can be realised in terms of polynomials satisfying a system of differential equations. As a consequence, the functions on which this particular class of operators act are functions taking values in the space of harmonics homogeneous of degree k. We prove the ellipticity of these operators and use this to investigate their kernel, focusing on polynomial solutions. Finally, we will also construct the fundamental solution using the theory of Riesz potentials.     

Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

     

On the product of projectors and generalized inverses

We consider generalized inverses of linear operators on arbitrary vector spaces and study the question when their product in reverse order is again a generalized inverse. This problem is equivalent to the question when the product of two projectors is again a projector, and we discuss necessary and sufficient conditions in terms of their kernels and images alone. We give a new representation of the product of generalized inverses that does not require explicit knowledge of the factors. Our approach is based on implicit representations of subspaces via their orthogonals in the dual space. For Fredholm operators, the corresponding computations reduce to finite-dimensional problems. We illustrate our results with examples for matrices and linear ordinary boundary problems.     

Cuntz�CKrieger Algebras and Wavelets on Fractals

We consider representations of Cuntz–Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron–Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets.     

A unified framework for harmonic analysis of functions on directed graphs and changing data

We present a general framework for studying harmonic analysis of functions in the settings of various emerging problems in the theory of diffusion geometry. The starting point of the now classical diffusion geometry approach is the construction of a kernel whose discretization leads to an undirected graph structure on an unstructured data set. We study the question of constructing such kernels for directed graph structures, and argue that our construction is essentially the only way to do so using discretizations of kernels. We then use our previous theory to develop harmonic analysis based on the singular value decomposition of the resulting non-self-adjoint operators associated with the directed graph. Next, we consider the question of how functions defined on one space evolve to another space in the paradigm of changing data sets recently introduced by Coifman and Hirn. While the approach of Coifman and Hirn requires that the points on one space should be in a known one-to-one correspondence with the points on the other, our approach allows the identification of only a subset of landmark points. We introduce a new definition of distance between points on two spaces, construct localized kernels based on the two spaces and certain interaction parameters, and study the evolution of smoothness of a function on one space to its lifting to the other space via the landmarks. We develop novel mathematical tools that enable us to study these seemingly different problems in a unified manner.     

Hilbert, Dirichlet and Fejér families of operators arising from C 0-groups, cosine functions and holomorphic semigroups

The main aim of this paper is to extend definitions of Hilbert transform, Dirichlet and Fejér operators (defined by convolution with suitable kernels in Lebesgue spaces) in arbitrary Banach spaces. We present a self-contained theory which includes different approaches of other authors whose starting points were usually C ₀-groups or cosine functions. We present relations with holomorphic semigroups. We characterize the geometric property of UMD spaces in terms of the Dirichlet and Fejér operators. To end the paper, we give examples to illustrate our results.     

The Superalgebra spl(p,q) and Differential Operators

Series of finite-dimensional representations of the superalgebrasspl(p,q) can be formulated in terms of linear differentialoperators acting on a suitable space of polynomials. We sketch the generalingredients necessary to construct these representations and presentexamples related to spl(2,1) and spl(2,2). By revisiting the products ofprojectivised representations of sl(2), we are able to construct new sets ofdifferential operators preserving some space of polynomials in two or morevariables. In particular, this allows us to express the representation ofspl(2,1) in terms of matrix differential operators in two variables. Thecorresponding operators provide the building blocks for the construction ofquasi-exactly solvable systems of two and four equations in two variables.We also present a quommutator deformation of spl(2,1) which, by constructionprovides an appropriate basis for analyzing the quasi exactly solvablesystems of finite difference equations.