Multiband sampling theorems for Mittag-Leffler transforms bandlimited on rays

We derive a biorthogonal Kramer analytic theorem, for integral transforms whose kernels generate biorthogonal bases in Hilbert spaces. The theorem is applied to various integral transforms associated with classes of fractional integro-differential eigenvalue problems, leading to Lagrange-type interpolation sampling theorems, derived by Djrabshian [Harmonic analysis and boundary value problems in the complex domain. Basel: Birkhäuser; 1993]. We work out some concrete examples, illustrating these sampling expansions.     

On Legendre functions of imaginary degree and associated integral transforms

New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function P_iξ(cosht). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ. A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler-Fok transforms, since conditions placed on functions are less restrictive.     

Describing functions: Atomic decompositions versus frames

The theory of frames and non-orthogonal series expansions with respect to coherent states is extended to a general class of spaces, the so-called coorbit spaces. Special cases include wavelet expansions for the Besov-Triebel-Lizorkin spaces, Gabortype expansions for modulation spaces, and sampling theorems for wavelet and Gabor transforms.     

Sampling Integrodifferential Transforms Arising from Second Order Differential Operators

We give sampling theorems associated with boundary value problems whose differential equations are of the form M(y) = λS(y), where M and S are differential expressions of the second and first order respectively and the eigenvalue parameter may appear in the boundary conditions. The class of the sampled functions is not a class of integral transforms as is the case in the classical sampling theory, but it is a class of integrodifferential transforms. We use solutions of the problem as well as Green's function to derive two sampling theorems.     

On sampling theory and basic Sturm–Liouville systems

We investigate the sampling theory associated with basic Sturm–Liouville eigenvalue problems. We derive two sampling theorems for integral transforms whose kernels are basic functions and the integral is of Jackson's type. The kernel in the first theorem is a solution of a basic difference equation and in the second one it is expressed in terms of basic Green's function of the basic Sturm–Liouville systems. Examples involving basic sine and cosine transforms are given.     

Changes of variables in modulation and Wiener amalgam spaces

In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of such spaces and, as a consequence, we obtain several versions of local and global Beurling–Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on ${\mathscr F}L^q$. Finally, counterparts of these results are discussed for spaces on the torus.     

Mellin-type differential equations and associated sampling expansions

This paper is devoted to a self-contained approach to Mellin-type differential equations and associated ssampling expansions. Here the first order differential operator is not the normal d/dx but D_M,c=xd/dx+c,c E R being connected with the definition of the Mellin transform. Existence and uniqueness theorems are established for a system of first order Mellin equations and the properties of nth order linear equations are investigated. Then self adjoint Mellin-type second order Sturm-Liouville eigenvalue problems are considered and properties of the eigenvalues, eigenfunctions and Green's functions are derived. As applications. sampling representations for two classes of integral transforms arising from the eigenvalue problem are introduced. In the first class the kernesl are solutions of the problem and in the second they are expressed in terms of green's function.     

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

We study an extension of the classical Paley–Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier–Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer’s expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier–Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.     

Triangular expansions

We use umbral methods to obtain several general expansion theorems for linear operators and linear functionals. We show in particular that every linear operator admits Newton-type expansions. These expansions are prototypes of numerous classical ones as well as new ones, some of which are powerful enough for use in numerical approximation.     

Transplantation Theorems��A Survey

In this survey article we overview transplantation theorems for several types of continuous and discrete orthogonal expansions. These include: Hankel and Dunkl transforms, and Fourier-Bessel, Jacobi and Laguerre expansions. We also discuss the idea of transference of transplantation and point out how a notion of conjugacy for orthogonal expansions may be interpreted as a generalized transplantation.     

Loaded differential operators: Convergence of spectral expansions

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a broad class of loaded even-order differential operators defined on a finite interval. These expansions are compared with the Fourier trigonometric series expansions of the same functions in an integral metric on any interior compact set of the main interval or on the entire interval. We obtain estimates for the equiconvergence rate of these expansions.     

Valuing continuous-installment options

Installment options are path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing European continuous-installment options written on dividend-paying assets in the standard Black–Scholes–Merton framework. The valuation of installment options can be formulated as a free boundary problem, due to the flexibility of continuing or stopping to pay installments. On the basis of a PDE for the initial premium, we derive an integral representation for the initial premium, being expressed as a difference of the corresponding European vanilla value and the expected present value of installment payments along the optimal stopping boundary. Applying the Laplace transform approach to this PDE, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form. Abelian theorems of Laplace transforms enable us to characterize asymptotic behaviors of the stopping boundary close and at infinite time to expiry. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the optimal stopping boundary.     

Integral Representations and Transforms of N-Functions. I

The author has proposed a new approach to extrapolation of operators from the scale of Lebesgue spaces to the Orlicz spaces beyond this scale. In this article comprising two parts we develop some mathematical method that enables us to prove extrapolation theorems for arbitrary behavior of an operator in the Lebesgue scale (i.e., in the case when the norm of the operator is an arbitrary function of p) and also in the case when the basic scale is an interval of the Lebesgue scale with exponents separated from 1 or +∞. In this event, we face ill-posed problems of inversion of the classical Mellin and Laplace type integral transforms over nonanalytic functions in terms of their asymptotic behavior on the real axis and also the question about the properties of convolution type integral transforms on classes of N-functions. In the first part of the article we study integral representations for N-functions by expansions in power functions with a positive weight and the behavior of convolution type integral transforms on classes of N-functions.     

Pointwise fourier inversion and related eigenfunction expansions

Necessary and sufficient conditions are found for the convergence at a pre-assigned point of the spherical partial sums (resp. integrals) of the Fourier series (resp. integral) in the class of piecewise smooth functions on Euclidean space. These results carry over unchanged to spherical harmonic expansions, Fourier transforms on hyperbolic space, and Dirichlet eigenfunction expansions with respect to the Laplace operator on a class of Riemannian manifolds. © 1994 John Wiley & Sons, Inc.     

Surgery of Spline-type and Molecular Frames

We prove a result about producing new frames for general spline-type spaces by piecing together portions of known frames. Using spline-type spaces as models for the range of certain integral transforms, we obtain results for time-frequency decompositions and sampling.     

Riesz–Jacobi Transforms as Principal Value Integrals

We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz–Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz–Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.     

Estimates of the local convergence rate of spectral expansions for even-order differential operators

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a wide class of even-order ordinary differential operators defined on a finite interval. These expansions are compared with the trigonometric Fourier series expansions of the same functions in the integral or uniform metric on an arbitrary interior compact set of the main interval as well as on the entire interval. We show the dependence of the equiconvergence rate of these expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the existence of infinitely many associated functions in the system of root functions.     

Sampling expansions associated with Kamke problems

The present paper is devoted to the derivation of sampling expansions for entire functions which are represented as integral transforms where a differential operator is acting on the kernels. The situation generalizes the results obtained in sampling theory associated with boundary value problems to the case when the differential equation has the form where N and P are two differential expressions of orders n and p respectively, and is the eigenvalue parameter. Both self adjoint and non self adjoint cases will be considered with examples in which the boundary conditions are strongly regular. Received February 16, 1998; in final form March 15, 1999     

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.     

Bilateral generating functions for Jacobi polynomials

In this paper, we have obtained three theorems on generating functions. We derive from these theorems a large number of bilateral generating functions for Jacobi polynomials. Certain interesting expansions of triple hypergeometric series are also obtained from one of the theorems.