General Orthogonal Polynomials Research Articles

DEK-type orthogonal polynomials and a modification of the Christoffel formula

In this note we revisit one of the first known examples of exceptional orthogonal polynomials that was introduced by Dubov, Eleonskii, and Kulagin in relation to nonharmonic oscillators with equidistant spectra. We dissect the DEK polynomials using the discrete Darboux transformations and unravel a characterization bypassing the differential equation that defines the DEK polynomials. This characterization also leads to a family of general orthogonal polynomials with missing degrees and this approach manifests its relation to biorthogonal polynomials introduced by Iserles and Nørsett, which are applicable to a whole range of problems in computational and applied analysis. We also obtain a modification of the Christoffel formula for this family since its classical form cannot be applied in this case.

Linearization and connection coefficients of polynomial sequences: A matrix approach

For any sequence of polynomials {pk(t)} in one real or complex variable, where pk has degree k for k≥0, we find explicit expressions and recurrence relations for infinite matrices whose entries are the numbers d(n,m,k), called linearization coefficients, that satisfypn(t)pm(t)=∑k=0n+md(n,m,k)pk(t),n,m≥0. For any pair of polynomial sequences {uk(t)} and {pk(t)} we find infinite matrices whose entries are the numbers e(n,m,k) that satisfypn(t)pm(t)=∑k=0n+me(n,m,k)uk(t),n,m≥0. We also obtain recurrence relations and other properties of the linearization coefficients. Such results are obtained using only simple algebraic properties of infinite matrices. We apply the general results to general orthogonal polynomial sequences and to some simple families of orthogonal polynomials, such as the Chebyshev, Hermite, and Charlier families.

A necessary and sufficient condition for convergence of the zeros of random polynomials

Consider random polynomials of the form Gn=∑i=0nξipi, where the ξi are i.i.d. non-degenerate complex random variables, and {pi} is a sequence of orthonormal polynomials with respect to a regular measure τ supported on a compact set K. We show that the zero measure of Gn converges weakly almost surely to the equilibrium measure of K if and only if Elog⁡(1+|ξ0|)<∞. This generalizes the corresponding result of Ibragimov and Zaporozhets in the case when pi(z)=zi. We also show that the zero measure of Gn converges weakly in probability to the equilibrium measure of K if and only if P(|ξ0|>en)=o(n−1).Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in Lp(τ), where p∈(0,∞]. In particular, sequences of Lp-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.

Dimension reduction for k-power bilinear systems using orthogonal polynomials and Arnoldi algorithm

In this paper, a dimension reduction method via general orthogonal polynomials and multiorder Arnoldi algorithm is proposed, which focuses on the topic of structure-preserving for k-power bilinear systems. The main procedure is using a series of expansion coefficient vectors of each state variables in the space spanned by general orthogonal polynomials that satisfy a recurrence formula to generate a projection based on multiorder Arnoldi algorithm. The resulting reduced-order model not only matches a desired number of expansion coefficients of the original output but also retains the topology structure. Meanwhile, the stability is well preserved under some certain conditions and the error bound is also given. Finally, two numerical simulations are provided to illustrate the effectiveness of our proposed algorithm in the views of accuracy and computational cost.

Time domain model reduction of time-delay systems via orthogonal polynomial expansions

This paper investigates model reduction of time-delay systems in the time domain based on the expansion of systems over orthogonal polynomials. Time-delay systems are first expanded over generalized Laguerre polynomials. The nice properties of generalized Laguerre polynomials lead to a direct system expansion and a Sylvester equation with special structures which enables an efficient calculation of Laguerre coefficients of systems. Projection methods are then adopted to generate reduced models, and we show that a desired number of Laguerre coefficients are preserved by reduced models. Further, we extend the proposed method to general orthogonal polynomials, where the relationship between Taylor expansion and orthogonal polynomial expansion is examined to achieve the expansion of time-delay systems. Systems with multiple delays and delays appearing in the derivative of the states are also discussed. Two numerical examples are simulated to showcase the efficiency of our approach.

Traveling wave analysis of rotating functionally graded graphene platelet reinforced nanocomposite cylindrical shells with general boundary conditions

This paper studies traveling wave motions of rotating multi-layered functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical shell under general boundary conditions. Theoretical equations are obtained according to Donnell shell theory, and artificial spring technique, where centrifugal and Coriolis effects caused by rotation are considered. By employing general orthogonal polynomials using a Gram-Schmidt process as admissible functions, solutions are achieved via Rayleigh-Ritz approach. Then, the accuracy and convergence of solutions are validated by the comparison of the obtained results with those reported in literature. Finally, free vibrations of FG-GPLRC cylindrical shells in both stationary and rotating states are investigated. The influences of boundary spring stiffness, GPL weigh fraction, total layer number, and geometry parameters on shell vibration characteristics are evaluated. It is revealed that the frequency variation trends along with material and geometric parameters are consistent for different boundary conditions, while variation rates and frequency values are highly dependent on boundary spring stiffness.

Structure‐preserved MOR method for coupled systems via orthogonal polynomials and Arnoldi algorithm

This study focuses on the topic of model order reduction (MOR) for coupled systems with inhomogeneous initial conditions and presents an MOR method by general orthogonal polynomials with Arnoldi algorithm. The main procedure is to use a series of expansion coefficients vectors in the space spanned by orthogonal polynomials that satisfy a recursive formula to generate a projection based on the multiorder Arnoldi algorithm. The resulting model not only match desired number of expansion coefficients but also has the same coupled structure as the original system. Moreover, the stability is preserved as well. The error bound between the outputs is well-discussed. Finally, numerical results show that the authors’ method can deal well with those systems with inhomogeneous initial conditions in the views of accuracy and computational cost.

Matrix biorthogonal polynomials on the real line: Geronimus transformations

In this paper, Geronimus transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasi-definite matrix of bivariate generalized functions with a well-defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The results are mainly concerned with the derivation of Christoffel-type formulas, which allow to express the perturbed matrix biorthogonal polynomials and its norms in terms of the original ones. The basic tool is the Gauss–Borel factorization of the Gram matrix, and particular attention is paid to the non-associative character, in general, of the product of semi-infinite matrices. The Geronimus transformation in which a right multiplication by the inverse of a matrix polynomial and an addition of adequate masses are performed, is considered. The resolvent matrix and connection formulas are given. Two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. This approach requires the perturbing matrix polynomial to have a nonsingular leading term. Then, using spectral techniques and spectral jets, Christoffel–Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques, that is valid also for singular leading coefficients. When the leading term is nonsingular, a comparison of between both methods is presented. The nonspectral method is applied to unimodular Christoffel perturbations, and a simple example for a degree one massless Geronimus perturbation is given.

Askey–Wilson polynomials and a double $q$-series transformation formula with twelve parameters

The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based on a $q$-series transformation formula and the Nassrallah--Rahman integral we prove a $q$--beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $q$-beta integral also allows us to derive a curious double $q$--series transformation formula, which includes one formula of Al--Salam and Ismail as a special case

Bessel summation inequalities for stability analysis of discrete‐time systems with time‐varying delays

SummaryThis paper is concerned with the stability analysis problems of discrete‐time systems with time‐varying delays using summation inequalities. In the literature focusing on the Lyapunov‐Krasovskii approach, the Jensen integral/summation inequalities have played important roles to develop less conservative stability criteria and thus have been widely studied. Recently, the Jensen integral inequality was successfully generalized to the Bessel‐Legendre inequalities constructed with arbitrary‐order Legendre polynomials. It was also shown that general inequality contributes to the less conservatism of stability criteria. In the case of discrete‐time systems, however, the Jensen summation inequality are hardly extensible to general ones since there have still not been general discrete orthogonal polynomials applicable to the developments of summation inequalities. Motivated by such observations, this paper proposes novel discrete orthogonal polynomials and then successfully derives general summation inequalities. The resulting summation inequalities are discrete‐time counterparts of the Bessel‐Legendre inequalities but are not based on the discrete Legendre polynomials. By developing hierarchical stability criteria based on the proposed summation inequalities, the effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of discrete‐time systems with time‐varying delays.

Matrix biorthogonal polynomials on the real line: Geronimus transformations

In this paper Geronimus transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasidefinite matrix of bivariate generalized functions with a well defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The results are mainly concerned with the derivation of Christoffel type formulas, which allow to express the perturbed matrix biorthogonal polynomials and its norms in terms of the original ones. The basic tool is the Gauss–Borel factorization of the Gram matrix, and particular attention is paid to the non-associative character, in general, of the product of semi-infinite matrices. The Geronimus transformation, in where a right multiplication by the inverse of a matrix polynomial and an addition of adequate masses is performed, is considered. The resolvent matrix and connection formulas are given. Two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. This approach requires the perturbing matrix polynomial to have a nonsingular leading term. Then, using spectral techniques and spectral jets, Christoffel–Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques, that is valid also for singular leading coefficients. When the leading term is nonsingular a comparative of between both methods is presented. The nonspectral method is applied to unimodular Christoffel perturbations, and a simple example for a degree one massless Geronimus perturbation is given.

A connection between time domain model order reduction and moment matching for LTI systems

ABSTRACTWe investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [1] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation, we show a connection to a rational Krylov subspace, and thus to moment matching. This theoretical link between the MOR techniques is illustrated by three numerical examples. For linear time-invariant systems, the link also motivates that the time-domain approach can be at best as accurate as moment matching, since the expansion points are fixed by the choice of the polynomial basis, while in moment matching they can be adapted to the system.

Arnoldi-based model order reduction for linear systems with inhomogeneous initial conditions

In this paper, we discuss the Arnoldi-based model order reduction (MOR) methods for linear systems with inhomogeneous initial conditions, and present a time domain MOR method by general orthogonal polynomials. The basic procedure is to use the expansion coefficient matrix in the orthogonal polynomial space satisfying a simple recurrence formula to generate a projection matrix, which is produced by a modified Arnoldi algorithm. Then, the resulting reduced model matches a desired number of the expansion coefficients of the original system. The approximate error estimate of the reduced model is given. Since the initial conditions are well represented by the subspaces constructed by our algorithm, it can well deal with those systems with inhomogeneous initial conditions. Two benchmark examples in real applications are simulated to illustrate the effectiveness of the proposed method.

Relative asymptotics for general orthogonal polynomials

We study right limits of the Bergman Shift matrix. Our results have applications to ratio asymptotics, weak asymptotic measures, relative asymptotics, and zero counting measures of the orthogonal and orthonormal polynomials.

Nevanlinna theory of the Wilson divided-difference operator

Sitting at the top level of the Askey-scheme, Wilson polynomials are regarded as the most general hypergeometric orthogonal polynomials. Instead of a differential equation, they satisfy a second order Sturm-Liouville type difference equation in terms of the Wilson divided-difference operator. This suggests that in order to better understand the distinctive properties of Wilson polynomials and related topics, one should use a function theory that is more natural with respect to the Wilson operator. Inspired by the recent work of Halburd and Korhonen, we establish a full-fledged Nevanlinna theory of the Wilson operator for meromorphic functions of finite order. In particular, we prove a Wilson analogue of the lemma on logarithmic derivatives, which helps us to derive Wilson operator versions of Nevanlinna's Second Fundamental Theorem, some defect relations and Picard's Theorem. These allow us to gain new insights on the distributions of zeros and poles of functions related to the Wilson operator, which is different from the classical viewpoint. We have also obtained a relevant five-value theorem and Clunie type theorem as applications of our theory, as well as a pointwise estimate of the logarithmic Wilson difference, which yields new estimates to the growth of meromorphic solutions to some Wilson difference equations and Wilson interpolation equations.

Derivative-Extended Time Domain Reduction for Coupled Systems Using Chebyshev Expansion

The time domain model reduction based on general orthogonal polynomials has been presented for linear systems. In this paper, we extend this approach by taking the derivative information of the system into account in the context of model reduction of coupled systems. We expand the derivative terms over the Chebyshev polynomial basis and show that Chebyshev coefficients of the expansion possess a specific structure, making it possible to preserve much more time domain information by employing projection methods. Besides, with the well-defined projection matrices, the resulting reduced model shares the same topological structure with the original coupled system. Two numerical examples are simulated to showcase the accuracy of incorporating the derivative information into model reduction.

A Connection between Time Domain Model Order Reduction and Moment Matching

AbstractWe investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [4] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation we show a connection to a rational Krylov subspace, and thus moment matching. This theoretical link between the MOR techniques is illustrated by one numerical example. (© 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Evaluation of spline functions for digital filtering problems

Methods for the evaluation of spline functions for digital filtering in data processing systems are developed. Basis polynomials of general form and basis discrete orthogonal polynomials are considered. Computations are organized by solving constrained optimization problems. Recurrences for the system of normalized discrete orthogonal polynomials and their derivatives are obtained. The proposed spline functions on discrete orthogonal polynomials reduce the computational cost and approximation errors compared with the case of general polynomials. The results of the statistical simulation of the application of spline functions based on discrete orthogonal polynomials in digital filtering problems are presented.

Model order reduction of MIMO bilinear systems by multi-order Arnoldi method

In this paper, we present a time domain model order reduction method for multi-input multi-output (MIMO) bilinear systems by general orthogonal polynomials. The proposed method is based on a multi-order Arnoldi algorithm applied to construct the projection matrix. The resulting reduced model can match a desired number of expansion coefficient terms of the original system. The approximate error estimate of the reduced model is given. And we also briefly discuss the stability preservation of the reduced model in some cases. Additionally, in combination with Krylov subspace methods, we propose a two-sided projection method to generate reduced models which capture properties of the original system in the time and frequency domain simultaneously. The effectiveness of the proposed methods is demonstrated by two numerical examples.

Multidimensional generalization of the Korous inequality

ABSTRACTThis paper deals with some qualitative properties of orthogonal polynomials in several variables. The boundedness and relations between two sets of orthonormal polynomials associated with an arbitrary weight function and its extension are investigated. It presents an analogy to Korous' result for general orthogonal polynomials in one variable.