Sequence Of Orthogonal Polynomials Research Articles

Energy-to-Peak State Estimation for Static Neural Networks With Interval Time-Varying Delays.

This paper is concerned with energy-to-peak state estimation on static neural networks (SNNs) with interval time-varying delays. The objective is to design suitable delay-dependent state estimators such that the peak value of the estimation error state can be minimized for all disturbances with bounded energy. Note that the Lyapunov-Krasovskii functional (LKF) method plus proper integral inequalities provides a powerful tool in stability analysis and state estimation of delayed NNs. The main contribution of this paper lies in three points: 1) the relationship between two integral inequalities based on orthogonal and nonorthogonal polynomial sequences is disclosed. It is proven that the second-order Bessel-Legendre inequality (BLI), which is based on an orthogonal polynomial sequence, outperforms the second-order integral inequality recently established based on a nonorthogonal polynomial sequence; 2) the LKF method together with the second-order BLI is employed to derive some novel sufficient conditions such that the resulting estimation error system is globally asymptotically stable with desirable energy-to-peak performance, in which two types of time-varying delays are considered, allowing its derivative information is partly known or totally unknown; and 3) a linear-matrix-inequality-based approach is presented to design energy-to-peak state estimators for SNNs with two types of time-varying delays, whose efficiency is demonstrated via two widely studied numerical examples.

Recurrence relations for the connection coefficients of classical orthogonal polynomials

ABSTRACTIn this paper, we present a framework on deriving recurrence relations for the connection coefficients of orthogonal polynomials. Let and be two sequences of classical orthogonal polynomials. Assume that there exists a non-trivial linear operator L such that and are also orthogonal polynomials. We show that the connection coefficients between and satisfy a linear recurrence of order at most 4. Based on the hypergeometric representation of orthogonal polynomials, this recurrence relation can be computed by the aid of the extended Zeilberger algorithm. We illustrate our method by several examples.

Characterization of the symmetric D-Laguerre–Hahn orthogonal polynomial sequences of even class via the quadratic decomposition

A regular form w is said to be D-Laguerre–Hahn of class the nonnegative integer , if there exist four coprime polynomials (monic), B, C and D (D depends on ) such that the Stieltjes function S(w) satisfies the Riccati differential equation . The class is then equal to . We proceed to the quadratic decomposition of symmetric D-Laguerre–Hahn orthogonal polynomial sequences of even class and we characterize the two derived sequences and their corresponding components u and v obtained by this approach. We describe the class and the Riccati equation of u and v into detail. Some examples from the class two are highlighted. In addition, some D-Laguerre–Hahn forms of class one are discovered.

Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials $$\{x^N\}_{N=0}^\infty $$ . We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.

Alternation points and bivariate Lagrange interpolation

Given m + 1 strictly decreasing numbers h 0 , h 1 , … , h m , we give an algorithm to construct a corresponding finite sequence of orthogonal polynomials p 0 , p 1 , … , p m such that p 0 = 1 , p j has degree j and p m − j ( h n ) = ( − 1 ) n p j ( h n ) for all j , n = 0 , 1 , … , m . Using these polynomials, we construct bivariate Lagrange polynomials and cubature formulas for nodes that are points in R 2 where the coordinates are taken from given finite decreasing sequences of the same length and where the indices have the same (or opposite) parity.

Julia Sets of Orthogonal Polynomials

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {P n } to properties of the support. More precisely we relate the Julia set of P n to the outer boundary of the support, the filled Julia set to the polynomial convex hull K of the support, and the Green’s function associated with P n to the Green’s function for the complement of K.

Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices

Let {c j } =0 be a sequence of matrix moments associated with a matrix of measures supported on the unit circle, and let {P j } =0 be its corresponding sequence of monic matrix orthogonal polynomials. In this contribution, we consider a perturbation on the moments and find an explicit relation for the perturbed orthogonal polynomials in terms of {P j } =0 . We also obtain an expression for the corresponding second kind polynomials.

Cubic decomposition of a family of semiclassical orthogonal polynomials of class two

ABSTRACTWe deal with a family of semiclassical orthogonal polynomial sequences of class two having the cubic decomposition , . Only four monic orthogonal polynomial sequences (MOPS) appear in which their recurrence coefficients are explicitly given.

Chebyshev polynomials of the second kind via raising operator preserving the orthogonality

It is well known that monic orthogonal polynomial sequences $$\{T_n\}_{n\ge 0}$$ and $$\{U_n\}_{n\ge 0}$$ , the Chebyshev polynomials of the first and second kind, satisfy the relation $$DT_{n+1}=(n+1)U_n$$ ( $$n\ge 0$$ ). One can also easily check that the following “inverse” of the mentioned formula holds: $${\mathcal {U}}_{-1}(U_n)=(n+1)T_{n+1}$$ ( $$n\ge 0$$ ), where $${\mathcal {U}}_\xi =x(xD+{\mathbb {I}})+\xi D$$ with $$\xi $$ being an arbitrary nonzero parameter and $${\mathbb {I}}$$ representing the identity operator. Note that whereas the first expression involves the operator D which lowers the degree by one, the second one involves $${\mathcal {U}}_\xi $$ which raises the degree by one (i.e. it is a “raising operator”). In this paper it is shown that the scaled Chebyshev polynomial sequence $$\{a^{-n}U_n(ax)\}_{n\ge 0}$$ where $$a^2=-\xi ^{-1}$$ , is actually the only monic orthogonal polynomial sequence which is $${\mathcal {U}}_\xi $$ -classical (i.e. for which the application of the raising operator $${\mathcal {U}}_\xi $$ turns the original sequence into another orthogonal one).

On Chebsyshev Solution of Curves by Using Gaussian Curvature

Gaussian curvature is commonly seen in the study of differential geometry. Gaussian curvature of a surface at a point is the product of the principal curvatures. They measure how the surface bends by different amounts in different directions at the point. Also, Gaussian curvature is given as the determinant of shape operator. In pure mathematics, differential equations are studied from different viewpoints. There are a lot of methods for solving differential equations in mathematics. From the differential equations viewpoint, Gaussian curvature solves the differential equation to find the main curve. One of them is Chebsyshev expansion method by using Chebsyshev polynomials. Also, they are important study in approximation theory. Chebyshev polynomials are a sequence of orthogonal polynomials and compose a polynomial sequence.The series solution is also used in surface of revolution. A surface of revolution is a surface generated by rotating a two-dimensional curve. In this study, our aim is to find the main curve by using Gaussian curvature. We substitute solution into the differential equation to find a relation for coeeficients of system. So, we use Chebsyshev polynomials for solutions to determine the curve and demonstrate our results on some well-known surfaces such as sphere, catenoid and torus.

Inner bounds for the extreme zeros of 3F2 hypergeometric polynomials

ABSTRACTZeilberger's celebrated algorithm finds pure recurrence relations (w.r.t. a single variable) for hypergeometric sums automatically. However, in the theory of orthogonal polynomials and special functions, contiguous relations w.r.t. several variables exist in abundance. We modify Zeilberger's algorithm to generate unknown contiguous relations that are necessary to obtain inner bounds for the extreme zeros of orthogonal polynomial sequences with hypergeometric representations. Using this method, we improve previously obtained upper bounds for the smallest and lower bounds for the largest zeros of the Hahn polynomials and we identify inner bounds for the extreme zeros of the Continuous Hahn and Continuous Dual Hahn polynomials. Numerical examples are provided to illustrate the quality of the new bounds. Without the use of computer algebra such results are not accessible. We expect our algorithm to be useful to compute useful and new contiguous relations for other hypergeometric functions.

Orthogonal polynomial approximation method for stability prediction in milling

Based on orthogonal polynomial approximation scheme, this paper presents several stability prediction methods using different kinds of orthogonal polynomials. The milling dynamics with consideration of the regenerative effect is described by time periodic delay-differential equations (DDEs). Firstly, this work employs the classical Legendre and Chebyshev polynomials to approximate the state term, delayed term, and periodic-coefficient matrix. With the help of direct integration scheme (DIS), the state transition matrixes which indicate the mapping relations of the dynamic response between the current tooth pass and the previous tooth pass are obtained. The stability lobe diagrams for single degree of freedom (DOF) and two DOF milling models are generated by using the Legendre and Chebyshev polynomial approximation-based methods. The rate of convergence of the Legendre and Chebyshev polynomial-based methods is compared with that of the benchmark first-order semi-discretization method (1stSDM). The comparison results indicate that the rate of convergence and the numerical stability of the Legendre and Chebyshev polynomial-based methods are both need to be improved. In order to develop new methods with high rate of convergence and numerical stability base on DIS, the monic orthogonal polynomial sequences are constructed by using Gram-Schmidt orthogonalization to approximate the state term, delayed term, and periodic-coefficient matrix. The rate of convergence and the computational efficiency of the monic orthogonal polynomial-based methods are evaluated by comparing with those of the benchmark 1stSDM. The results turn out that the monic orthogonal polynomial-based methods are advantageous in terms of the rate of convergence and numerical stability. The stability lobe diagrams for single DOF and two DOF milling models obtained by the monic orthogonal polynomial-based methods are compared with those obtained by the 1stSDM. Finally, the monic orthogonal polynomial-based methods are proved to be the effective and efficient methods to predict the milling stability.

Polynomial sequences generated by infinite Hessenberg matrices

Abstract We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.

Orthogonal polynomials associated with an inverse spectral transform. The cubic case

The purpose of this work is to give some new algebraic properties of the orthogonality of a monic polynomial sequence {Qn}n ? o defined by Qn(X) = Pn(X) + SnPn-1(X) + tnPn-2(X) + rnPn-3(X), n ? 1, where rn ? 0, n ? 3, and {Pn}n?0 is a given sequence of monic orthogonal polynomials. Essentially, we consider some cases in which the parameters rn, sn, and tn can be computed more easily. Also, as a consequence, a matrix interpretation using LU and UL factorization is done. Some applications for Laguerre, Bessel and Tchebychev orthogonal polynomials of second kind are obtained.

Nonsymmetric Askey–Wilson polynomials and Q-polynomial distance-regular graphs

In his famous theorem (1982), Douglas Leonard characterized the q-Racah polynomials and their relatives in the Askey scheme from the duality property of Q-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the q-Racah polynomials in the above situation. Let Γ denote a Q-polynomial distance-regular graph that contains a Delsarte clique C. Assume that Γ has q-Racah type. Fix a vertex x∈C. We partition the vertex set of Γ according to the path-length distance to both x and C. The linear span of the characteristic vectors corresponding to the cells in this partition has an irreducible module structure for the universal double affine Hecke algebra Hˆq of type (C1∨,C1). From this module, we naturally obtain a finite sequence of orthogonal Laurent polynomials. We prove the orthogonality relations for these polynomials, using the Hˆq-module and the theory of Leonard systems. Changing Hˆq by Hˆq−1 we show how our Laurent polynomials are related to the nonsymmetric Askey–Wilson polynomials, and therefore how our Laurent polynomials can be viewed as nonsymmetric q-Racah polynomials.

The CMV Bispectral Problem

A classical result due to Bochner classifies the orthogonal polynomials on the real line which are common eigenfunctions of a second order linear differential operator. We settle a natural version of the Bochner problem on the unit circle which answers a similar question concerning orthogonal Laurent polynomials and can be formulated as a bispectral problem involving CMV matrices. We solve this CMV bispectral problem in great generality proving that, except the Lebesgue measure, no other one on the unit circle yields a sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear differential operator of arbitrary order. Actually, we prove that this is the case even if such an eigenfunction condition is imposed up to finitely many orthogonal Laurent polynomials.

Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings

Let $\mu$ be a probability measure with an infinite compact support on $\mathbb{R}$. Let us further assume that $(F_n)_{n=1}^\infty$ is a sequence of orthogonal polynomials for $\mu$ where $(f_n)_{n=1}^\infty$ is a sequence of nonlinear polynomials and $F_n:=f_n\circ\dots\circ f_1$ for all $n\in\mathbb{N}$. We prove that if there is an $s_0\in\mathbb{N}$ such that $0$ is a root of $f_n^\prime$ for each $n>s_0$ then the distance between any two zeros of an orthogonal polynomial for $\mu$ of a given degree greater than $1$ has a lower bound in terms of the distance between the set of critical points and the set of zeros of some $F_k$. Using this, we find sharp bounds from below and above for the infimum of distances between the consecutive zeros of orthogonal polynomials for singular continuous measures.

Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas

When a nontrivial measure μ on the unit circle satisfies the symmetry dμ(ei(2π−θ))=−dμ(eiθ) then the associated orthogonal polynomials on the unit circle, say Φn, are all real. In this case, in 1986, Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zΦn(z)+Φn⁎(z)} and {zΦn(z)−Φn⁎(z)}, where Φn⁎(z)=znΦn(1/z‾)‾, satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [−1,1]. The same authors, in 1988, have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently the extension associated with the para-orthogonal polynomials zΦn(z)−Φn⁎(z) was thoroughly explored, especially from the point of view of three term recurrence and chain sequences. The main objective of the present article is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zΦn(z)+Φn⁎(z) for any nontrivial measure on the unit circle. As an important application of the theory, a characterization for the existence of the integral ∫02π|eiθ−w|−2dμ(eiθ), where w is such that |w|=1, is given in terms of the coefficients αn−1=−Φn(0)‾, n≥1. Examples are also provided to justify all the results.

A characterization of some semi classical orthogonal polynomials

We show that if is a sequence of monic orthogonal polynomials satisfying a differential-difference equation of the formthen the orthogonal polynomial sequence is necessarily orthogonal to a well known semi classical generalization of the classical Hermite weight. If is allowed to be any real number then we show that the Laguerre polynomials also appear as a limiting case. The method is based on a derivation of a pair of non linear difference equations which are a special case of the asymmetric discrete Painleve IV equations. Such equations appear entirely as a consequence of the differential-difference equation together with the three term recurrence relation for a general sequence of orthogonal polynomials as opposed to previous derivations that stem from the consideration of a specific weight function.

Appell polynomial sequences with respect to some differential operators

We present a study of a specific kind of lowering operator, herein called $\Lambda$, which is defined as a finite sum of lowering operators, proving that this configuration can be altered, for instance, by the use of Stirling numbers. We characterize the polynomial sequences fulfilling an Appell relation with respect to $\Lambda$, and considering a concrete cubic decomposition of a simple Appell sequence, we prove that the polynomial component sequences are $\Lambda$-Appell, with $\Lambda$ defined as previously, although by a three term sum. Ultimately, we prove the non-existence of orthogonal polynomial sequences which are also $\Lambda$-Appell, when $\Lambda$ is the lowering operator $\Lambda=a_{0}D+a_{1}DxD+a_{2}\left(Dx\right)^2D$, where $a_{0}$, $a_{1}$ and $a_{2}$ are constants and $a_{2} \neq 0$. The case where $a_{2}=0$ and $a_{1} \neq 0$ is also naturally recaptured.