scholarly journals A set of orthogonal polynomials induced by a given orthogonal polynomial

1993 ◽  
Vol 46 (1-2) ◽  
pp. 174-198 ◽  
Author(s):  
Walter Gautschi ◽  
Shikang Li
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials

An Application of Separate Convergence for Continued Fractions to Orthogonal Polynomials

1992 ◽  
Vol 35 (3) ◽  
pp. 381-389
Author(s):  
William B. Jones ◽  
W. J. Thron ◽  
Nancy J. Wyshinski
Keyword(s):  
Distribution Function ◽  
Asymptotic Formula ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Continued Fractions ◽  
Polynomial Sequence ◽  
Convergence Results ◽  
Compact Subsets

AbstractIt is known that the n-th denominators Qn (α, β, z) of a real J-fractionwhereform an orthogonal polynomial sequence (OPS) with respect to a distribution function ψ(t) on ℝ. We use separate convergence results for continued fractions to prove the asymptotic formulathe convergence being uniform on compact subsets of

scholarly journals A characterization of the four Chebyshev orthogonal families

2005 ◽  
Vol 2005 (13) ◽  
pp. 2071-2079 ◽  
Author(s):  
E. Berriochoa ◽  
A. Cachafeiro ◽  
J. M. Garcia-Amor
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Linear Combinations ◽  
Classical Orthogonal Polynomial ◽  
Polynomial Sequences ◽  
Fixed Length ◽  
Constant Coefficients ◽  
Fourth Kind

We obtain a property which characterizes the Chebyshev orthogonal polynomials of first, second, third, and fourth kind. Indeed, we prove that the four Chebyshev sequences are the unique classical orthogonal polynomial families such that their linear combinations, with fixed length and constant coefficients, can be orthogonal polynomial sequences.

Application of Generalized Orthogonal Polynomials to Parameter Estimation of Time-Invariant and Bilinear Systems

1987 ◽  
Vol 109 (1) ◽  
pp. 7-13 ◽  
Author(s):  
Maw-Ling Wang ◽  
Shwu-Yien Yang ◽  
Rong-Yeu Chang
Keyword(s):  
Parameter Estimation ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Computational Algorithm ◽  
State Equation ◽  
Dynamic State ◽  
Generalized Orthogonal Polynomials ◽  
Generalized Orthogonal ◽  
Time Invariant ◽  
Estimation Of Time

Generalized orthogonal polynomials (GOP) which can represent all types of orthogonal polynomials and nonorthogonal Taylor series are first introduced to estimate the parameters of a dynamic state equation. The integration operation matrix (IOP) and the differentiation operation matrix (DOP) of the GOP are derived. The key idea of deriving IOP and DOP of these polynomials is that any orthogonal polynomial can be expressed by a power series and vice versa. By employing the IOP approach to the identification of time invariant systems, algorithms for computation which are effective, simple and straightforward compared to other orthogonal polynomial approximations can be obtained. The main advantage of using the differentiation operation matrix is that the parameter estimation can be carried out starting at an arbitrary time of interest. In addition, the computational algorithm is even simpler than that of the integral operation matrix. Illustrative examples for using IOP and DOP approaches are given. Very satisfactory results are obtained.

scholarly journals Comparison of the Orthogonal Polynomial Solutions for Fractional Integral Equations

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 59
Author(s):  
Ayşegül Daşcıoğlu ◽  
Serpil Salınan
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Integral Equations ◽  
Collocation Method ◽  
Fractional Integral ◽  
The Other ◽  
Numerical Examples ◽  
Polynomial Solutions

In this paper, a collocation method based on the orthogonal polynomials is presented to solve the fractional integral equations. Six numerical examples are given to illustrate the method. The results are compared with the other methods in the literature, and the results obtained by different kinds of polynomials are compared.

Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations

1982 ◽  
Vol 25 (3) ◽  
pp. 291-295 ◽  
Author(s):  
Lance L. Littlejohn ◽  
Samuel D. Shore
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Image Position

AbstractOne of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation:In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.

A PROPAGATING ALGORITHM FOR DETERMINING NTH-ORDER POLYNOMIAL, LEAST‐SQUARES FITS

Geophysics ◽  
1977 ◽  
Vol 42 (6) ◽  
pp. 1265-1276 ◽  
Author(s):  
Anthony F. Gangi ◽  
James N. Shapiro
Keyword(s):  
Experimental Data ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Least Squares ◽  
High Precision ◽  
Lower Order ◽  
Hankel Matrix ◽  
Polynomial Fit ◽  
Order Polynomial ◽  
Data Points

An algorithm is described which iteratively solves for the coefficients of successively higher‐order, least‐squares polynomial fits in terms of the results for the previous, lower‐order polynomial fit. The technique takes advantage of the special properties of the least‐squares or Hankel matrix, for which [Formula: see text]. Only the first and last column vectors of the inverse matrix are needed at each stage to continue the iteration to the next higher stage. An analogous procedure may be used to determine the inverse of such least‐squares type matrices. The inverse of each square submatrix is determined from the inverse of the previous, lower‐order submatrix. The results using this algorithm are compared with the method of fitting orthogonal polynomials to data points. While the latter method gives higher accuracy when high‐order polynomials are fitted to the data, it requires many more computations. The increased accuracy of the orthogonal‐polynomial fit is valuable when high precision of fitting is required; however, for experimental data with inherent inaccuracies, the added computations outweigh the possible benefit derived from the more accurate fitting. A Fortran listing of the algorithm is given.

A Proceedure for Systematically Fitting Local Volume Functions Using Orthogonal Polynomials

1970 ◽  
Vol 46 (3) ◽  
pp. 225-228
Author(s):  
L. C. Wensel ◽  
J. Van Roessel
Keyword(s):  
Computer Program ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Polynomial Equations ◽  
Computational Process ◽  
End Points ◽  
Local Volume ◽  
Best Fit ◽  
Volume Functions

The proposed technique is designed to fit orthogonal polynomial equations to existing local volume tables. This method has the advantage of being able to find the "best" fit of up to 8 powers (terms) such that (1) the function will yield volumes that increase with increasing diameter, and (2) will keep the end points of the curves within range. The computer program is designed so that it can be used with little or no knowledge of the computational process involved.

scholarly journals A new proof of a theorem about generalized orthogonal polynomials

1991 ◽  
Vol 14 (2) ◽  
pp. 393-397
Author(s):  
A. McD. Mercer
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Recent Result ◽  
Asymptotic Distribution ◽  
Generalized Polynomials ◽  
Generalized Orthogonal Polynomials ◽  
Generalized Orthogonal ◽  
Polynomial Theory

In this note it is shown that a fairly recent result on the asymptotic distribution of the zeros of generalized polynomials can be deduced from an old theorem ofG. Polya, using a minimum of orthogonal polynomial theory.

On Linear Ordinary Differential Equation With Orthogonal Polynomial Solutions

2018 ◽  
Vol 1 (20) ◽  
pp. 559-572
Author(s):  
ولدان وليد محمود
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Differential Operators ◽  
Linear Ordinary Differential Equation ◽  
Large Piece ◽  
Polynomial Solutions ◽  
The Subject

: The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. In this paper we give a  survey of the orthogonal polynomial solutions of second-order and fourth-order linear ordinary differential equation, on the generated self-adjoint differential operators .

scholarly journals Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise

2018 ◽  
Vol 21 (02) ◽  
pp. 1850011
Author(s):  
Eugene Lytvynov ◽  
Irina Rodionova
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Fock Space ◽  
Linear Span ◽  
Representation Formula ◽  
Linear Operators ◽  
Lévy Noise ◽  
Meixner Class ◽  
Polynomial Formula ◽  
Levy Noise

Let [Formula: see text] denote a non-commutative monotone Lévy process. Let [Formula: see text] denote the corresponding monotone Lévy noise, i.e. formally [Formula: see text]. A continuous polynomial of [Formula: see text] is an element of the corresponding non-commutative [Formula: see text]-space [Formula: see text] that has the form [Formula: see text], where [Formula: see text]. We denote by [Formula: see text] the space of all continuous polynomials of [Formula: see text]. For [Formula: see text], the orthogonal polynomial [Formula: see text] is defined as the orthogonal projection of the monomial [Formula: see text] onto the subspace of [Formula: see text] that is orthogonal to all continuous polynomials of [Formula: see text] of order [Formula: see text]. We denote by [Formula: see text] the linear span of the orthogonal polynomials. Each orthogonal polynomial [Formula: see text] depends only on the restriction of the function [Formula: see text] to the set [Formula: see text]. The orthogonal polynomials allow us to construct a unitary operator [Formula: see text], where [Formula: see text] is an extended monotone Fock space. Thus, we may think of the monotone noise [Formula: see text] as a distribution of linear operators acting in [Formula: see text]. We say that the orthogonal polynomials belong to the Meixner class if [Formula: see text]. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: [Formula: see text] and [Formula: see text]. In this case, the monotone Lévy noise has the representation [Formula: see text]. Here, [Formula: see text] and [Formula: see text] are the (formal) creation and annihilation operators at [Formula: see text] acting in [Formula: see text].

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