Optimally Space Localized Polynomials with Applications in Signal Processing

For the filtering of peaks in periodic signals, we specify polynomial filters that are optimally localized in space. The space localization of functions f having an expansion in terms of orthogonal polynomials is thereby measured by a generalized mean value ε(f). Solving an optimization problem including the functional ε(f), we determine those polynomials out of a polynomial space that are optimally localized. We give explicit formulas for these optimally space localized polynomials and determine in the case of the Jacobi polynomials the relation of the functional ε(f) to the position variance of a well-known uncertainty principle. Further, we will consider the Hermite polynomials as an example on how to get optimally space localized polynomials in a non-compact setting. Finally, we investigate how the obtained optimal polynomials can be applied as filters in signal processing.     

Divided Differences and Linearly Recursive Sequences

We show that the theory of divided differences is a natural tool for the study of linearly recurrent sequences. The divided differences functional associated with a monic polynomial w on degree n + 1 yields a vector space isomorphism between the space of polynomials of degree at most equal to n and the space of linearly recurrent sequences f that satisfy the difference equation w(E)f=0 where E is the usual shift operator. Using such isomorphisms, we can translate problems about recurrent sequences into simple problems about polynomials. We present here a new approach to the theory of divided differences, using only generating functions and elementary linear algebra, which clarifies the connections of divided differences with rational functions, polynomial interpolation, residues, and partial fractions decompositions.     

Computing ECT-B-splines recursively

ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as Neville–Aitken weights of certain generalized interpolation problems. For multiple knots they are limits of Neville–Aitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval [a,b] connected at inner knots all of multiplicities zero by full connection matrices A ^[i] that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECT-spline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves. *Supported in part by INTAS 03-51-6637.     

A two-dimensional matrix Padé-type approximation in the inner product space

By introducing a bivariate matrix-valued linear functional on the scalar polynomial space, a general two-dimensional (2-D) matrix Padé-type approximant (BMPTA) in the inner product space is defined in this paper. The coefficients of its denominator polynomials are determined by taking the direct inner product of matrices. The remainder formula is developed and an algorithm for the numerator polynomials is presented when the generating polynomials are given in advance. By means of the Hankel-like coefficient matrix, a determinantal expression of BMPTA is presented. Moreover, to avoid the computation of the determinants, two efficient recursive algorithms are proposed. At the end the method of BMPTA is applied to partial realization problems of 2-D linear systems.     

Numerical solution of some three-dimensional problems of elasticity theory by Vekua's method

We consider the solution of the problem of elastic equilibrium of a three-dimensional orthotropic plate in the absence of displacements on the end surfaces under the action of forces applied to the lateral surfaces. The solution of the original problem by Vekua's method is reduced to the solution of a recursive sequence of two-dimensional problems. A numerical solution of these problems is obtained by computer using the finite-difference method. The effect of the number of Legendre polynomials on the accuracy with which the boundary conditions are satisfied is investigated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 77–84, 1986.     

Coherent states for a generalized oscillator with a finite-dimensional Hilbert space

A construction of oscillator-like systems connected with a given set of orthogonal polynomials and coherent states for such systems developed by the authors is extended to the case of systems with a finite-dimensional state space. As an example, we consider a generalized oscillator connected with Krawtchouk polynomials. Bibliography: 24 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 75–99.     

Recursive Three-Term Recurrence Relations for?the?Jacobi Polynomials on a Triangle

Given a suitable weight on ℝ ^d , there exist many (recursive) three-term recurrence relations for the corresponding multivariate orthogonal polynomials. In principle, these can be obtained by calculating pseudoinverses of a sequence of matrices. Here we give an explicit recursive three-term recurrence for the multivariate Jacobi polynomials on a simplex. This formula was obtained by seeking the best possible three-term recurrence. It defines corresponding linear maps, which have the same symmetries as the spaces of Jacobi polynomials on which they are defined. The key idea behind this formula is that some Jacobi polynomials on a simplex can be viewed as univariate Jacobi polynomials, and for these the recurrence reduces to the univariate three-term recurrence.     

Computation of the canonical polynomials and applications to some optimal control problems

The Tau method is a numerical technique that consists in constructing polynomial approximate solutions for ordinary differential equations. This method has two approaches: operational and recursive. The former converts the differential problem to a matrix problem and produces approximations in terms of a prescribed orthogonal polynomials basis. In the recursive approach, we construct approximate solutions in terms of a special set of polynomials {Q _k (t); k?=?0, 1, 2...} called canonical polynomials basis. In some cases, the Q _k ??s can be obtained explicitly through a recursive formula. But no analogous formulae are reported in the literature for the general cases. In this paper, utilizing the operational Tau method, we develop an algorithm that allows to generate those canonical polynomials iteratively and explicitly. In addition, we demonstrate the capability of the operational Tau method in treating quadratic optimal control problems governed by ordinary differential equations.     

On the Laguerre-Hahn Intertwining Operator and Application to Connection Formulas

In this paper, we show that the lowering operator D _u indexed by a linear functional on polynomials u, introduced by F. Marcellán and R. Sfaxi, namely the Laguerre-Hahn derivative, is intertwining with the standard derivative D by a linear isomorphism S _u on polynomials. This allows us to establish an intertwining relation between the nonsingular Laguerre-Hahn polynomials of class zero of Hermite type and the Hermite polynomials, as well as some new connection formulas between such Laguerre-Hahn polynomials and the canonical basis.     

Transpose-free Lanczos-type algorithms for nonsymmetric linear systems

The method of Lanczos for solving systems of linear equations is implemented by using recurrence relationships between formal orthogonal polynomials. A drawback is that the computation of the coefficients of these recurrence relationships usually requires the use of the transpose of the matrix of the system. Due to the indirect addressing, this is a costly operation. In this paper, a new procedure for computing these coefficients is proposed. It is based on the recursive computation of the products of polynomials appearing in their expressions and it does not involve the transpose of the matrix. Moreover, our approach allows to implement simultaneously and at a low price a Lanczos-type product method such as the CGS or the BiCGSTAB. This revised version was published online in June 2006 with corrections to the Cover Date.     

Perturbations on the subdiagonals of Toeplitz matrices

Let L be an Hermitian linear functional defined on the linear space of Laurent polynomials. It is very well known that the Gram matrix of the associated bilinear functional in the linear space of polynomials is a Toeplitz matrix. In this contribution we analyze some linear spectral transforms of L such that the corresponding Toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial Toeplitz matrix. We focus our attention in the analysis of the quasi-definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one.     

A new computational approach for stochastic fluid models of multiplexers with heterogeneous sources

In this paper we derive an efficient computational procedure for the system in which fluid is produced byN ₁ on-off sources of type 1,N ₂ on-off sources of type 2 and transferred to a buffer which is serviced by a channel of constant capacity. This is a canonical model for multiservice ATM multiplexing, which is hard to analyze and also of wide interest. This paper's approach to the computation of the buffer overflow probability,G(x) = Pr{buffer content >x}, departs from all prior approaches in that it transforms the computation ofG(x) for a particularx into a recursive construction of an interpolating polynomial. For the particular case of two source types the interpolating polynomial is in two variables. Our main result is the derivation of recursive algorithms for computing the overflow probabilityG(x) and various other performance measures using their respective relations to two-dimensional interpolating polynomials. To make the computational procedure efficient we first derive a new system of equations for the coefficients in the spectral expansion formula forG(x) and then use specific properties of the new system for efficient recursive construction of the polynomials. We also develop an approximate method with low complexity and analyze its accuracy by numerical studies. We computeG(x) for different values ofx, the mean buffer content and the coefficient of the dominant exponential term in the spectral expansion ofG(x). The accuracy of the approximations is reasonable when the buffer utilization characterized by G(0) is more than 10^–2.     

Quantum Schur algebras revisited

A geometric construction of quantum Schur algebras was given by Beilinson, Lusztig and MacPherson in terms of pairs of flags in a vector space. By viewing such pairs of flags as representations of a poset, we give a recursive formula for the structure constants of quantum Schur algebras which is related to certain Hall polynomials. As an application, we provide a direct proof of the fundamental multiplication formulas which play a key role in the Beilinson-Lusztig-MacPherson realization of quantum gl_n. In the appendix we show how to groupoidify quantum Schur algebras in the sense of Baez, Hoffnung and Walker.     

Products of B-patches

Products and tensor products of multivariate polynomials in B-patch form are viewed as linear combinations of higher degree B-patches. Univariate B-spline segments and certain regions of simplex splines are examples of B-patches. A recursive scheme for transforming tensor product B-patch representations into B-patch representations of more variables is presented. The scheme can also be applied for transforming ann-fold product of B-patch expansions into a B-patch expansion of higher degree. Degree raising formulas are obtained as special cases. The scheme calculates the blossom of the (tensor) product surface and generalizes the pyramidal recursive scheme for B-patches.     

Improving projection‐based eigensolvers via adaptive techniques

We consider subspace iteration (or projection‐based) algorithms for computing those eigenvalues (and associated eigenvectors) of a Hermitian matrix that lie in a prescribed interval. For the case that the projector is approximated with polynomials, we present an adaptive strategy for selecting the degree of these polynomials such that convergence is achieved with near‐to‐optimum overall work without detailed a priori knowledge about the eigenvalue distribution. The idea is then transferred to the approximation of the projector by numerical integration, which corresponds to FEAST algorithm proposed by E. Polizzi in 2009. [E. Polizzi: Density‐matrix‐based algorithm for solving eigenvalue problems. Phys. Rev. B 2009; 79 :115112]. Here, our adaptation controls the number of integration nodes. We also discuss the interaction of the method with search space reduction methods.     

An extension theorem for slice monogenic functions and some of its consequences

Slice monogenic functions were introduced by the authors in [6]. The central result of this paper is an extension theorem, which shows that every holomorphic function defined on a suitable domain D of a complex plane can be uniquely extended to a slice monogenic function defined on a domain U _D , determined by D, in a Euclidean space of appropriate dimension. Two important consequences of the result are a structure theorem for the zero set of a slice monogenic function (with a related corollary for polynomials with coefficients in Clifford algebras), and the possibility to construct a multiplicative theory for such functions. Slice monogenic functions have a very important application in the definition of a functional calculus for n-tuples of noncommuting operators.     

Immanant positivity for Catalan-Stieltjes matrices

We give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to produce inequalities for many combinatorial polynomials, such as the Eulerian polynomials, Schröder polynomials, and Narayana polynomials.     

Automorphisms of the solution spaces of special double-confluent Heun equations

Two new linear operators determining automorphisms of the solution space of a special double-confluent Heun equation in the general case are obtained. This equation has two singular points, both of which are irregular. The obtained result is applied to solve the nonlinear equation of the resistively shunted junction model for an overdamped Josephson junction in superconductors. The new operators are explicitly expressed in terms of structural polynomials, for which recursive computational algorithms are constructed. Two functional equations for the solutions of the special double-confluent Heun equation are found.