Ph. Delsarte

The Nevanlinna–Pick Problem for Matrix-Valued Functions

This paper contains a detailed treatment of the Nevanlinna–Pick interpolation problem for matrix-valued functions. A close relationship with the trigonometric moment problem is put into light. Matrix extensions of Pick’s solvability criterion and Nevanlinna’s iterative algorithm are presented. Necessary and sufficient conditions are obtained for uniqueness of the solution of the infinite interpolation problem. The approach is essentially based upon the theories of J-contractive transformations and of Weyl matrix circles.

Bounds for systems of lines, and Jacobi polynomials

Bounds are obtained for the cardinality of sets of lines having a prescribed number of angles, both in real and in complex Euclidean n-space. Extremal sets provide combinatorial configurations with a particular algebraic structure, such as association schemes and regular two-graphs. The bounds are derived by use of matrix techniques and the addition formula for Jacobi polynomials.

Schur Parametrization of Positive Definite Block-Toeplitz Systems

The parameters occurring in Szego’s recurrence relations associated with a class C function are known to be the same as the Schur parameters of the corresponding class S function. The present paper contains a matrix extension of this result. Certain important questions about the matrix classes S and C, related to spectral factorization, are studied by means of their Schur–Szego parameters.

Fisher type inequalities for Euclidean t-designs

The notion of a Euclidean t-design is analyzed in the framework of appropriate inner product spaces of polynomial functions. Some Fisher type inequalities are obtained in a simple manner by this method. The same approach is used to deal with certain analogous combinatorial designs.

A method of matrix inverse triangular decomposition based on contiguous principal submatrices

Abstract An algorithm is presented which performs the triangular decomposition of the inverse of a given matrix. The method is applicable to any matrix all contiguous principal submatrices of which are nonsingular. The algorithm is particularly efficient when the matrix has certain partial symmetries exhibited by the Toeplitz structure.

Discrete Prolate Spheroidal Wave Functions and Interpolation

We describe an algorithm for the interpolation of burst errors in discrete-time signals that can be modelled as being band-limited. The algorithm correctly restores a mutilated signal that is indeed band-limited. The behavior of the algorithm when applied to signals containing noise or out-of-band components can be analysed satisfactorily with the aid of asymptotic properties of the discrete prolate spheroidal sequences and wave functions. The effect of windowing can also be described conveniently in terms of these sequences and functions.

Parametric toeplitz systems

This paper investigates the properties of the two-variable polynomialu (λ, z) built on the first column of the adjoint matrix ofλI -C, whereC is a given Hermitian Toeplitz matrix. In particular, the stability properties ofu (λ,z) are discussed and are shown to depend essentially on the location of X with respect to the eigenvalues ofC. The eigenvectors ofC, which have recently found some applications in signal processing and estimation theory, are obtained from the polynomialu(λ,z) whenλ tends to the eigenvalues ofC. This allows one to derive several results concerning the eigenpolynomials, including those for the case of multiple eigenvalues.

On the Toeplitz embedding of an arbitrary matrix

Abstract It has been shown by Delosme and Morf that an arbitrary block matrix can be embedded into a block Toeplitz matrix; the dimension of this embedding depends on the complexity of the matrix structure compared to the block Toeplitz structure. Due to the special form of the embedding matrix, the algebra of matrix polynomials relative to block Toeplitz matrices can be interpreted directly in terms of the original matrix and therefore can be extended to arbitrary matrices. In fact, these polynomials turn out to provide an appropriate framework for the recently proposed generalized Levinson algorithm solving the general matrix inversion problem.

An equivalence relation in planar least squares inverse approximation

It is shown that the Planar Least Square Inverse (PLSI) of degree (sm, tn) relative to the lacunary polynomial b(z<inf>1</inf>^s, z<<inf>2</inf>^t)is obtained from the PLSI of degree (m, n) relative to b(z<inf>1</inf>, z<inf>2</inf>) by the substitution z<inf>1</inf>→ z<inf>1</inf>^s, z<inf>2</inf>→ z<<inf>2</inf>^t.

Automorphism Groups of Convolutional Codes

Let K be the monomial group of degree n, over the field