arXiv subjects

Tomas Sauer

Publications and source records attributed to Tomas Sauer.

24 records · Page 2Linked to original sources

arXiv2017

Observations on interpolation by total degree polynomials in two variables

In contrast to the univariate case, interpolation with polynomials of a given maximal total degree is not always possible even if the number of interpolation points and the space dimension coincide. Due to that, numerous constructions for interpolation sets have been devised, the most popular ones being based on intersections of lines. In this paper, we study algebraic properties of some such interpolation configurations, namely the approaches by Radon-Berzolari and Chung-Yao. By means of proper H-bases for the vanishing ideal of the configuration, we derive properties of the matrix of first syzygies of this ideal which allow us to draw conclusions on the geometry of the point configuration.

math.NA↗

arXiv2016

Prony's method in several variables

The paper gives an extension of Prony's method to the multivariate case which is based on the relationship between polynomial interpolation, normal forms modulo ideals and H--bases.

math.NA↗

arXiv2014

Decomposition of tensors

We consider representations of tensors as sums of decomposable tensors or, equivalently, decomposition of multilinear forms into one--forms. In this short note we show that there exists a particular finite strongly orthogonal decomposition which is essentially unique and yields all critical points of the multilinear form on the torus. In particular, this determines exactly the number of critical points of the multilinear form, giving an affirmative answer to a finiteness conjecture by Friedland.

math.NA↗

arXiv2014

Kernels of discrete convolutions and subdivision operators

We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in connected directly though somewhat intricately to the multiplicity of the common zeros of certain multivariate polynomials, a concept introduced by Gröbner in the description of kernels of partial differential operators with constant coefficients. These results can are then used to determine the kernels of stationary subdivision operators as well.

math.NA↗

arXiv2013

Hermite subdivision schemes, exponential polynomial generation, and annihilators

We consider the question when the so--called spectral condition} for Hermite subdivision schemes extends to spaces generated by polynomials and exponential functions. The main tool are convolution operators that annihilate the space in question which apparently is a general concept in the study of various types of subdivision operators. Based on these annihilators, we characterize the spectral condition in terms of factorization of the subdivision operator.

math.NA↗

arXiv2007

Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis

In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of non-stationary bivariate subdivision schemes, which allow to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition.

math.NA↗