Orthogonal Rational Functions Research Articles

Identification of Nonlinear Systems Using the Hammerstein-Wiener Model with Improved Orthogonal Functions

Hammerstein-Wiener systems present a structure consisting of three serial cascade blocks. Two are static nonlinearities, which can be described with nonlinear functions. The third block represents a linear dynamic component placed between the first two blocks. Some of the common linear model structures include a rational-type transfer function, orthogonal rational functions (ORF), finite impulse response (FIR), autoregressive with extra input (ARX), autoregressive moving average with exogenous inputs model (ARMAX), and output-error (O-E) model structure. This paper presents a new structure, and a new improvement is proposed, which is consisted of the basic structure of Hammerstein-Wiener models with an improved orthogonal function of Müntz-Legendre type. We present an extension of generalised Malmquist polynomials that represent Müntz polynomials. Also, a detailed mathematical background for performing improved almost orthogonal polynomials, in combination with Hammerstein-Wiener models, is proposed. The proposed approach is used to identify the strongly nonlinear hydraulic system via the transfer function. To compare the results obtained, well-known orthogonal functions of the Legendre, Chebyshev, and Laguerre types are exploited.

Complex Jacobi matrices generated by Darboux transformations

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number, including the properties of the corresponding sequences of orthogonal polynomials. We also present some invariant and semi-invariant properties of Jacobi matrices under such transformations. For instance, we show that a Nevai class is invariant under the transformations in question, which is not true in general, and that the ratio asymptotic still holds outside the spectrum of the corresponding symmetric complex Jacobi matrix but the spectrum could include one extra point. In principal, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to RII-recurrence relations, which in turn are related to orthogonal rational functions and pencils of Jacobi matrices.

Orthogonal rational functions with real poles, root asymptotics, and GMP matrices

There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on R \mathbb {R} and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for ∞ \infty . We extend aspects of this theory in the setting of rational functions with poles on R ¯ = R ∪ { ∞ } \overline {\mathbb {R}} = \mathbb {R} \cup \{\infty \} , obtaining a formulation which allows multiple poles and proving an invariance with respect to R ¯ \overline {\mathbb {R}} -preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.

Pole Identification using discrete Laguerre expansion and variable projection

We propose a novel algorithm for identifying the poles of transfer functions describing SISO-LTI (single input single output, linear time invariant) systems. Our Identification method works in the frequency domain and consists of two parts. In the first part, we extend a discrete Laguerre expansion based method with an automatic parameter selection scheme. This allows us to find an initial estimate of the poles of SISO-LTI transfer functions without the need for human intuition. Then, in the second part, we propose a novel optimization problem to improve our initial estimates. The proposed optimization aims to reduce the least squared error of a parameterized model, which can be interpreted as an orthogonal projection of the system's frequency response onto a subspace spanned by Generalized Orthogonal Rational Basis functions (GOBFs). We solve the corresponding nonlinear optimization task using gradient based methods, where we can analytically calculate the gradient of the error functional. Through robust numerical experiments, we investigate the behavior of the developed methods and show that they work even in scenarios, when the transfer function has a high number of poles.

Orthogonal rational functions arising from the Chebyshev polynomials of first and second kind

We develop sequences of orthogonal rational functions arising from rational function mappings applied to the Classical Chebyshev polynomials of the first and second kind. Explicit representations for the rational functions and their associated norms are derived. The work in this manuscript provides the basis for a general framework for rational function mappings akin to the polynomial mapping framework developed by Geronimo and Van Assche [10].

Orthogonal Cauchy-like matrices

Cauchy-like matrices arise often as building blocks in decomposition formulas and fast algorithms for various displacement-structured matrices. A complete characterization for orthogonal Cauchy-like matrices is given here. In particular, we show that orthogonal Cauchy-like matrices correspond to eigenvector matrices of certain symmetric matrices related to the solution of secular equations. Moreover, the construction of orthogonal Cauchy-like matrices is related to that of orthogonal rational functions with variable poles.

Generation of orthogonal rational functions by procedures for structured matrices

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

Sobolev orthogonal Legendre rational spectral methods for exterior problems

The purpose of this paper is to develop the diagonalized Legendre rational spectral method for exterior problems. We first consider the exterior problems of two-dimensional elliptic and parabolic equations in polar coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. Then we consider the exterior problems of three-dimensional elliptic and parabolic equations in spherical coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. The main advantages of the suggested approaches are that the discrete systems are diagonal and the numerical solutions can be represented as truncated Fourier series. The numerical results show their effectiveness and accuracy.

The convergence properties of orthogonal rational functions on the extended real line and analytic on the upper half plane

We study the orthogonal rational functions with a given sequence of poles on the half plane. We give the weak-star convergence results for the rational measures which relate to the Poisson kernel and reproducing kernel. Those rational measures represent the same inner product as the normalized Nevanlinna measure in [Formula: see text]. Moreover, we consider the convergence properties of the interpolants which interpolate the Carathéodory function [Formula: see text].

Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula

Abstract We prove a bijective unitary correspondence between (1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum ${\textsf{E}}$ and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as ${\textsf{E}}$-dependent operator Möbius transforms of certain generating CMV matrices that are periodic up to a rotational phase; for this reason we call them “MCMV.” Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated to such CMV matrices arise as quadratic irrationalities.

A Family of Orthogonal Rational Functions and Other Orthogonal Systems with a skew-Hermitian Differentiation Matrix

In this paper we explore orthogonal systems in mathrm {L}_2(mathbb {R}) which give rise to a skew-Hermitian, tridiagonal differentiation matrix. Surprisingly, allowing the differentiation matrix to be complex leads to a particular family of rational orthogonal functions with favourable properties: they form an orthonormal basis for mathrm {L}_2(mathbb {R}), have a simple explicit formulae as rational functions, can be manipulated easily and the expansion coefficients are equal to classical Fourier coefficients of a modified function, hence can be calculated rapidly. We show that this family of functions is essentially the only orthonormal basis possessing a differentiation matrix of the above form and whose coefficients are equal to classical Fourier coefficients of a modified function though a monotone, differentiable change of variables. Examples of other orthogonal bases with skew-Hermitian, tridiagonal differentiation matrices are discussed as well.

Resolvent Matrix of the Truncated Nevanlinna-Pick Matrix Interpolation Problem via Orthogonal Rational Functions in the Stieltjes Class

Resolvent Matrix of the Truncated Nevanlinna-Pick Matrix Interpolation Problem via Orthogonal Rational Functions in the Stieltjes Class

Orthogonal rational functions on the extended real line and analytic on the upper half plane

Let $\{\alpha _k\}_{k=1}^\infty$ be an arbitrary sequence of complex numbers in the upper half plane. We generalize the orthogonal rational functions $\phi _n$ based upon those points and obtain the Nevanlinna measure, together with the Riesz and Poisson kernels, for Caratheodory functions $F(z)$ on the upper half plane. Then, we study the relation between ORFs and their functions of the second kind as well as their interpolation properties. Further, by using a linear transformation, we generate a new class of rational functions and state the necessary conditions for guaranteeing their orthogonality.

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these ORF when the poles are all outside or all inside the unit disk, or when they can be anywhere in the extended complex plane outside the unit circle. Some properties of matrices that are the product of elementary unitary transformations will be proved and some connections with related algorithms for direct and inverse eigenvalue problems will be explained.

A note on Wall’s modification of the Schur algorithm and linear pencils of Jacobi matrices

In this note we revive a transformation that was introduced by H. S. Wall and that establishes a one-to-one correspondence between continued fraction representations of Schur, Carathéodory, and Nevanlinna functions. This transformation can be considered as an analog of the Szegő mapping but it is based on the Cayley transform, which relates the upper half-plane to the unit disk. For example, it will be shown that, when applying the Wall transformation, instead of OPRL, we get a sequence of orthogonal rational functions that satisfy three-term recurrence relation of the form (H−λJ)u=0, where u is a semi-infinite vector, whose entries are the rational functions. Besides, J and H are Hermitian Jacobi matrices for which a version of the Denisov–Rakhmanov theorem holds true. Finally, we will demonstrate how pseudo-Jacobi polynomials (aka Routh–Romanovski polynomials) fit into the picture.

Diophantine properties of orthogonal polynomials and rational functions

Diophantine properties of orthogonal polynomials and rational functions

A comparison of numerical and semi-analytical methods for the case of heat transfer equations arising in porous medium

Natural convective heat transfer in porous media which is of importance in the design of canisters for nuclear waste disposal has received considerable attention during the past few decades. This paper presents a comparison between two different analytical and numerical methods, i.e. pseudospectral and Adomian decomposition methods. The pseudospectral approach makes use of the orthogonal rational Jacobi functions; this method reduces the solution of the problem to a solution of a system of algebraic equations. Numerical results are compared with each other, showing that the pseudospectral method leads to more accurate results and is applicable on similar problems.

Factorial-type Schur functions, orthogonal rational functions, and discrete dressing chains

A relationship between orthogonal rational functions and discrete integrable systems is studied by an approach based on Schur-type symmetric functions. A system of orthogonal rational functions is constructed using a multiparameter deformation of the Schur functions. Spectral equations for the orthogonal rational functions are derived by using properties of the Schur-type symmetric functions. The compatibility condition of the spectral equations induces a discrete dressing chain which is a Toda-type discrete integrable system describing dressing transformations for orthogonal rational functions.

On three-term recurrence and Christoffel–Darboux identity for orthogonal rational functions on the real line

ABSTRACTFirst, we give an algebraic proof to the Christoffel–Darboux identity of formal orthogonal rational functions on the real line by exposing some underlying algebraic properties. This proof does not involve the three-term recurrence relationship. Besides, it is shown that if a family of rational functions satisfies the Christoffel–Darboux relation, then it also admits a three-term recurrence relationship. Thus, the equivalence between both relations is revealed.

Fast Computing on Vehicle Dynamics Using Chebyshev Series Expansions

This paper focuses on faster computation techniques to integrate mechanical models in electronic advanced active safety applications. It shows the different techniques of approximation in series of functions and differential equations applied to vehicle dynamics. This allows the achievement of approximate polynomial and rational solutions with a very fast and efficient computation. First, the whole theoretical basic principles related to the techniques used are presented: orthogonality of functions, function expansion in Chebyshev and Jacobi series, approximation through rational functions, the Minimax–Remez algorithm, orthogonal rational functions, and the perturbation of dynamic systems theory, that reduces the degree of the expansion polynomials used. As an application, it is shown the obtaining of approximate solutions to the longitudinal dynamics, vertical dynamics, steering geometry, and a tyre model, all obtained through development in series of orthogonal functions with a computation much faster than those of its equivalents in the classic vehicle theory. These polynomial partially symbolic solutions present very low errors and very favorable analytical properties due to their simplicity, becoming ideal for real-time computation as those required for the simulation of evasive manoeuvres prior to a crash. This set of techniques had never been applied to vehicle dynamics before.