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We provide an algorithm for the construction of orthonormal multivariate polynomials that are symmetric with respect to the interchange of any two coordinates on the unit hypercube and are constrained to the hyperplane where the sum of the…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…
In many applications (hupergeometric-type) special functions like orthogonal polynomials are needed. For example in more than 50% of the published solutions for the (application-oriented) questions in the "Problems Section" of SIAM Review…
We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways of computing the correlation functions of a discrete orthogonal polynomial…
Finding a computationally efficient algorithm for the inverse continuous wavelet transform is a fundamental topic in applications. In this paper, we show the convergence of the inverse wavelet transform.
Computation of (approximate) polynomials common factors is an important problem in several fields of science, like control theory and signal processing. While the problem has been widely studied for scalar polynomials, the scientific…
I propose an orthogonalization procedure preserving the grading of the initial graded set of linearly independent vectors. In particular, this procedure is applicable for orthonormalization of any countable set of polynomials in several…
Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial $P$, one must first construct a corresponding complementary polynomial $Q$. Existing approaches to this…
We investigate different approaches to transform a given binomial into a monomial via blowing up appropriate centers. In particular, we develop explicit implementations in {\sc Singular}, which allow to make a comparison on the basis of…
Many authors studied numeric algorithms for solving the linear systems of the pentadiagonal type. The well-known Fast Pentadiagonal System Solver algorithm is an example of such algorithms. The current article are described new numeric and…
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…
Fourier transform of multivariate orthogonal polynomials on the unit ball are obtained. By using Parseval's identity, a new family of multivariate orthogonal functions are introduced. The results are expressed in terms of the continuous…
We propose an algorithm for quickly evaluating polynomials. It pre-conditions a complex polynomial $P$ of degree $d$ in time $O(d\log d)$, with a low multiplicative constant independent of the precision. Subsequent evaluations of $P$…
Rational function approximations find applications in many areas including macro-modeling of high-frequency circuits, model order reduction for controller design, interpolation and extrapolation of system responses, surrogate models for…
Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier…
Minimal annihilating polynomials are very useful in a wide variety of algorithms in exact linear algebra. A new efficient method is proposed for calculating the minimal annihilating polynomials for all the unit vectors, for a square matrix…
Recently, the butterfly approximation scheme and hierarchical approximations have been proposed for the efficient computation of integral transforms with oscillatory and with asymptotically smooth kernels. Combining both approaches, we…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
In this paper we introduce a new approach to the concept of multipolynomials and generalize several results of the homogeneous polynomials and symmetric multilinear applications. We also present an abstract approach to the concept of…
We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial…