A. S. Cavaretta

The design of curves and surfaces by subdivision algorithms

We present a survey of the basic principles and concepts associated with subdivision algorithms. Special attention is given to the convergence question for these algorithms, and the mathematical tools suitable for analysing convergence are presented.

A factorization theorem for banded matrices

In this paper we prove a factorization theorem for strictly m-banded totally positive matrices. We show that such a matrix is a product of m one-banded matrices with positive entries. A bi-infinite matrix A =( Ai, i), - ca i, and Ai i_n,Ai i #O. These matrices arise quite freqiently in applications, and recently the; have been the subject of several independent investigations [2,3]. In this paper we prove a factorization theorem for strictly m-handed totally positive matrices. We show that such a matrix is a product of m one-banded matrices with positive entries. Let us begin by recalling that a matrix A is totally positive if and only if ull its minors are nonnegative:

On the Solvability of Certain Systems of Linear Difference Equations

For a certain class of block Toeplitz matrices, we identify the smallest sector containing the zeros of the determinant for the corresponding symbol.

LACUNARY TRIGONOMETRIC INTERPOLATION ON EQUIDISTANT NODES

Publisher Summary This chapter discusses the interpolation at equidistant nodes of a given function and certain of its derivatives by trigonometric polynomials. The unique interpolation is possible only if even derivative values and odd derivative values are used in equal quantities. If n = 2r is even, the problem of trigonometric interpolation is regular on X precisely when oq – ep = 0 if q = 2p, and 1 if q = 2p+1. The chapter also discusses the case n odd: n = 2r+1 and explicit forms for the fundamental polynomials.

Pyramid patches provide potential polynomial paradigms

Recursive schemes are central to curve and surface representation by piecewise polynomials. Many algorithms for the evaluation, conversion, subdivision, differentiation, degree raising of piecewise polynomial curves and surfaces are given in recursive form. This has the advantage of providing computational simplicity and stability while at the same time revealing the beautiful geometric and algebraic structure of these representation forms. This paper is an exploratory presentation into the varied world of pyramidal schemes for the generation of multivariate polynomial surfaces. We focus primarily on lineal polynomials, that is, polynomials formed by products of linear forms. Through the multinomial theorem we are led to dual systems of polynomials whose properties are reflections of those possessed by lineal polynomials. Connections to multilinear symmetric forms, the multiparameter multinomial distribution and the theory of permanents is made.

Lipschitz-type bounds for the map A→|A| on L(H)

It is well known that the absolute value map on the self-adjoint operators on an infinite dimensional Hilbert spaces is not Lipschitz continuous, although Lipschitz continuity holds on certain subsets of operators. In this note, we provide an elementary proof that the absolute value map is Lipschitz continuous on the set of all operators which are bounded below in norm by any fixed positive constant. Applications are indicated.

Optimal recovery of interpolation operators in Hardy spaces

We continue studies begun by C.A. Micchelli and T.J. Rivlin on optimal recovery in Hp spaces. The feature operators are various interpolation operators drawn from the theory of Walsh equiconvergence, as are the information sets. The theory is of interest in that it identifies linear algorithms which might not otherwise be isolated for study or used as approximations of the feature operators. In some cases, we can identify the optimal algorithm although we cannot explicitly determine the exact order of the approximation that it achieves.

The volume of restricted moment spaces

In this paper we compute the volume of restricted moment spaces under very general conditions on the functions generating the moment space.