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We construct a commutative algebra Ax of difference operators in R p

In this paper we treat the two-variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. This problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes. We show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition. As a corollary of the main result a necessary and sufficient condition for the existence of a spectral Fejér-Riesz factorization of a strictly positive two-variable trigonometric polynomial is given in terms of the Fourier coefficients of its reciprocal.Tools in the proofs include a specific two-variable Kronecker theorem based on certain elements from algebraic geometry, as well as a two-variable Christoffel-Darboux like formula. The key ingredient is a matrix valued polynomial that appears in a parametrized version of the Schur-Cohn test for stability. The results also have consequences in the theory of two-variable orthogonal polynomials where a spectral matching result is obtained, as well as in the study of inverse formulas for doubly-indexed Toeplitz matrices. Finally, numerical results are presented for both the autoregressive filter problem and the factorization problem.

Fractal interpolation functions are used to construct a compactly supported continuous, orthogonal wavelet basis spanning L 2 (IR). The wavelets share many of the properties normally associated with spline wavelets in particular linear phase.

Abstract-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. In this paper, we propose a general algorithm to compute multiwavelet transform coeficients 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 discretetime vector-valued signals. The proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete-time signals. We 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.

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, Szegö’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.

ABSTRACT. Starting from a sequence {pn{x; no)} of orthogonal polynomials with an orthogonality measure yurj supported on Eo C [-1,1], we construct a new sequence {p"(x;fi)} of orthogonal polynomials on£ = T~1(Eq) (T is a polynomial of degree TV) with an orthogonality measure [i that is relatedwill in general consist of TV intervals. We give explicit formulas relating {pn(x;fi)} and {pn(x;fio)} and show how the recurrence coefficients in the three-term recurrence formulas for these orthogonal polynomials are related. If one chooses T to be a Chebyshev polynomial of the first kind, then one gets sieved orthogonal polynomials.

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