Jeffrey S. Geronimo

Design of prefilters for discrete multiwavelet transforms

The pyramid algorithm for computing single wavelet transform coefficients is well known. The pyramid algorithm can be implemented by using tree-structured multirate filter banks. The authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets. The proposed algorithm can be thought of as a discrete vector-valued wavelet transform for certain discrete-time vector-valued signals. The proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete-time signals. The authors then present some numerical experiments to illustrate the performance of the algorithm, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms.

Construction of Orthogonal Wavelets Using Fractal Interpolation Functions

Fractal interpolation functions are used to construct a compactly supported continuous, orthogonal wavelet basis spanning

Riemann-Hilbert Problems for Multiple Orthogonal Polynomials

L^2 (\mathbb{R})

SCATTERING THEORY AND POLYNOMIALS ORTHOGONAL ON THE REAL LINE

. The wavelets share many of the properties normally associated with spline wavelets, in particular, they have linear phase.

Orthogonal polynomials with asymptotically periodic recurrence coefficients

In the early nineties, Fokas, Its and Kitaev observed that there is a natural Riemann-Hilbert problem (for 2 x×2 matrix functions) associated with a system of orthogonal polynomials. This Riemann-Hilbert problem was later used by Deift et al. and Bleher and Its to obtain interesting results on orthogonal polynomials, in particular strong asymptotics which hold uniformly in the complex plane. In this paper we will show that a similar Riemann-Hilbert problem (for (r + 1) × (r + 1) matrix functions) is associated with multiple orthogonal polynomials. We show how this helps in understanding the relation between two types of multiple orthogonal polynomials and the higher order recurrence relations for these polynomials. Finally we indicate how an extremal problem for vector potentials is important for the normalization of the Riemann-Hilbert problem. This extremal problem also describes the zero behavior of the multiple orthogonal polynomials.

Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets

The techniques of scattering theory are used to study polynomials orthogonal on a segment of the real line. Instead of applying these techniques to the usual three-term recurrence formula, we derive a set of two two-term re- currence formulas satisfied by these polynomials. One of the advantages of these new recurrence formulas is that the Jost function is related, in the limit as n -» oo, to the solution of one of the recurrence formulas with the boundary conditions given at n = 0. In this paper we investigate the properties of the Jost function and the spectral function assuming the coefficients in the recurrence formulas converge at a particular rate. of the more familiar three-term recurrence formula satisfied by polynomials or- thogonal on the real line (PORL). These two two-term recurrence formulas have several interesting properties. For example, the Jost function, which has been shown to be so useful in the theory of orthogonal polynomials (4), (7), is the limit of a sequence of polynomials satisfying one of the recurrence formulas with the boundary condition given at n = 0. It is natural to ask whether such a system of recurrence relations exists for PORL. In this paper we develop the theory of PORL along a line that parallels the theory of POUC and delve deeper into the consequences of applying scattering theory to PORL. In §11 we define the polynomials and derive the familiar three-term recurrence formula. Now, in analogy with POUC, a set of two two-term recurrence formulas is derived. These formulas plus the appropriate boundary conditions are taken as the fundamental equations defining the polynomials. From them the Christoffel- Darboux formula and Wronskian theorem are derived. In §111 the Jost function is defined and is shown to be the limit of a sequence of polynomials satisfying one of the recurrence formulas with the boundary condi- tions given at n = 0. Some of the properties of the Jost function are investigated. Since we have started with the recurrence relations it is necessary to show that the polynomials are indeed orthogonal. This is done in §IV. We also show how one can calculate the Jost function directly from the weight function.

Scattering Theory and Matrix Orthogonal Polynomials on the Real Line

Abstract Given the coefficients in the three term recurrence relation satisfied by orthogonal polynomials, we investigate the properties of those classes of polynomials whose coefficients are asymptotically periodic. Assuming a rate of convergence of the coefficients to their asymptotic values, we construct the measure with respect to which the polynomials are orthogonal and discuss their asymptotic behavior.

Scattering theory and polynomials orthogonal on the unit circle

Orthogonal polynomials are used to construct families of C0 and C1 orthogonal, compactly supported spline multiwavelets. These families are indexed by an integer which represents the order of approximation. We indicate how to obtain from these multiwavelet bases for L2 [0,1] and present a C2 example.

Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform

The techniques of scattering and inverse scattering theory are used to investigate the properties of matrix orthogonal polynomials. The discrete matrix analog of the Jost function is introduced and its properties investigated. The matrix distribution function with respect to which the polynomials are orthonormal is constructed. The discrete matrix analog of the Marchenko equation is derived and used to obtain further results on the matrix Jost function and the distribution function.

An Exact Formula for the Measure Dimensions Associated with a Class of Piecewise Linear Maps

The techniques of scattering theory are used to investigate polynomials orthogonal on the unit circle. The discrete analog of the Jost function, which has been shown to play an important role in the theory of polynomials orthogonal on a segment of the real line, is defined for this system and its properties are investigated. The relation between the Jost function and the weight function is discussed. The techniques of inverse scattering theory are developed and used to obtain new asymptotic formulas satisfied by the polynomials. A set of sum rules satisfied by the coefficients in the recurrence relaxation is exhibited. Finally, Szego’s theorem on Toeplitz determinants is proved using the recurrence formulas and the Jost function. The techniques of inverse scattering theory are used to find the correction terms.