scholarly journals Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

2022 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Alejandro Molano
Keyword(s):  
Orthogonal Polynomials ◽  
Fourier Coefficients ◽  
Finite Interval ◽  
Laguerre Polynomials ◽  
Inner Product ◽  
Sobolev Type ◽  
Content Type ◽  
Constructive Methods ◽  
Reasonable Choice ◽  
Connection Formulas

Purpose In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.Design/methodology/approach To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.Findings Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.Originality/value In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

scholarly journals A numerical method for solving the Cauchy problem for ODEs using a system of polynomials generated by a system of modified Laguerre polynomials

2019 ◽  
pp. 13-24
Author(s):  
Gasan Akniyev ◽  
Ramis Gadzhimirzaev
Keyword(s):  
Cauchy Problem ◽  
Fourier Series ◽  
Approximate Calculation ◽  
Fourier Coefficients ◽  
Laguerre Polynomials ◽  
Numerical Realization ◽  
Inner Product ◽  
Sobolev Type ◽  
The Cauchy Problem ◽  
Modified Laguerre Polynomials

In this paper, we consider a numerical realization of an iterative method for solving the Cauchy problem for ordinary differential equations, based on representing the solution in the form of a Fourier series by the system of polynomials $\{L_{1,n}(x;b)\}_{n=0}^\infty$, orthonormal with respect to the Sobolev-type inner product $$ \langle f,g\rangle=f(0)g(0)+\int_{0}^\infty f'(x)g'(x)\rho(x;b)dx $$ and generated by the system of modified Laguerre polynomials $\{L_{n}(x;b)\}_{n=0}^\infty$, where $b>0$. In the approximate calculation of the Fourier coefficients of the desired solution, the Gauss -- Laguerre quadrature formula is used.

scholarly journals On Second Order q-Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1300
Author(s):  
Carlos Hermoso ◽  
Edmundo J. Huertas ◽  
Alberto Lastra ◽  
Anier Soria-Lorente
Keyword(s):  
Orthogonal Polynomials ◽  
Real Number ◽  
Difference Equations ◽  
Arbitrary Number ◽  
Recurrence Formula ◽  
Second Order ◽  
Inner Product ◽  
Sobolev Type ◽  
Ladder Operators ◽  
Connection Formulas

This contribution deals with the sequence {Un(a)(x;q,j)}n≥0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q∈(0,1). We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by Un(a)(x;q,j), which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality.

scholarly journals On the Uniform Convergence of the Fourier Series by the System of Polynomials Generated by the System of Laguerre Polynomials

2020 ◽  
Vol 20 (4) ◽  
pp. 416-423
Author(s):  
Ramis M. Gadzhimirzaev ◽  
Keyword(s):  
Fourier Series ◽  
Sobolev Space ◽  
Weight Function ◽  
Uniform Convergence ◽  
Laguerre Polynomials ◽  
Inner Product ◽  
Sobolev Type ◽  
Absolutely Continuous ◽  
Differentiable Functions ◽  
Proof Of Convergence

Let w(x) be the Laguerre weight function, 1 ≤ p < ∞, and Lpw be the space of functions f, p-th power of which is integrable with the weight function w(x) on the non-negative axis. For a given positive integer r, let denote by WrLpw the Sobolev space, which consists of r−1 times continuously differentiable functions f, for which the (r−1)-st derivative is absolutely continuous on an arbitrary segment [a, b] of non-negative axis, and the r-th derivative belongs to the space Lpw. In the case when p = 2 we introduce in the space WrL2w an inner product of Sobolev-type, which makes it a Hilbert space. Further, by lαr,n(x), where n = r, r + 1, ..., we denote the polynomials generated by the classical Laguerre polynomials. These polynomials together with functions lαr,n(x) = xn / n! , where n = 0, 1, r − 1, form a complete and orthonormal system in the space WrL2w. In this paper, the problem of uniform convergence on any segment [0,A] of the Fourier series by this system of polynomials to functions from the Sobolev space WrLpw is considered. Earlier, uniform convergence was established for the case p = 2. In this paper, it is proved that uniform convergence of the Fourier series takes place for p > 2 and does not occur for 1 ≤ p < 2. The proof of convergence is based on the fact that WrLpw ⊂ WrL2w for p > 2. The divergence of the Fourier series by the example of the function ecx using the asymptotic behavior of the Laguerre polynomials is established.

A higher order Sobolev-type inner product for orthogonal polynomials in several variables

2014 ◽  
Vol 68 (1) ◽  
pp. 35-46 ◽  
Author(s):  
Herbert Dueñas ◽  
Luis E. Garza ◽  
Miguel Piñar
Keyword(s):  
Orthogonal Polynomials ◽  
Higher Order ◽  
Inner Product ◽  
Sobolev Type ◽  
Several Variables

scholarly journals Orthogonal polynomials with varying weight of Laguerre type

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1053-1062 ◽  
Author(s):  
Predrag Rajkovic ◽  
Sladjana Marinkovic ◽  
Miomir Stankovic
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Laguerre Polynomials ◽  
Theoretical Physics ◽  
Inner Product ◽  
Exponential Functions ◽  
Polynomial Sequence ◽  
Functional Product ◽  
Real Polynomials ◽  
Differential Properties

In this paper, we define and examine a new functional product in the space of real polynomials. This product includes the weight function which depends on degrees of the participants. In spite of it does not have all properties of an inner product, we construct the sequence of orthogonal polynomials. These polynomials can be eigenfunctions of a differential equation what was used in some considerations in the theoretical physics. In special, we consider Laguerre type weight function and prove that the corresponding orthogonal polynomial sequence is connected with Laguerre polynomials. We study their differential properties and orthogonal properties of some related rational and exponential functions.

scholarly journals Neoteric formulas of the monic orthogonal Chebyshev polynomials of the sixth-kind involving moments and linearization formulas

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Waleed M. Abd-Elhameed ◽  
Youssri H. Youssri
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Connection Coefficients ◽  
Current Article ◽  
In Virtue Of ◽  
Connection Formulas

AbstractThe principal aim of the current article is to establish new formulas of Chebyshev polynomials of the sixth-kind. Two different approaches are followed to derive new connection formulas between these polynomials and some other orthogonal polynomials. The connection coefficients are expressed in terms of terminating hypergeometric functions of certain arguments; however, they can be reduced in some cases. New moment formulas of the sixth-kind Chebyshev polynomials are also established, and in virtue of such formulas, linearization formulas of these polynomials are developed.

scholarly journals Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients

2021 ◽  
pp. 1-35
Author(s):  
Nadiia Derevianko ◽  
Gerlind Plonka
Keyword(s):  
Rational Approximation ◽  
Fourier Coefficients ◽  
Exponential Sums ◽  
Finite Interval ◽  
Large Set ◽  
Recovery Method ◽  
Exact Reconstruction ◽  
Finite Set ◽  
Number Formula ◽  
Numerical Approaches

In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form [Formula: see text], where the frequency parameters [Formula: see text] are pairwise distinct. In order to reconstruct [Formula: see text] we employ a finite set of classical Fourier coefficients of [Formula: see text] with regard to a finite interval [Formula: see text] with [Formula: see text]. For our method, [Formula: see text] Fourier coefficients [Formula: see text] are sufficient to recover all parameters of [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The recovery is based on the observation that for [Formula: see text] the terms of [Formula: see text] possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA Algorithm for rational approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494A1522]. If a sufficiently large set of [Formula: see text] Fourier coefficients of [Formula: see text] is available (i.e. [Formula: see text]), then our recovery method automatically detects the number [Formula: see text] of terms of [Formula: see text], the multiplicities [Formula: see text] for [Formula: see text], as well as all parameters [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], [Formula: see text], determining [Formula: see text]. Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

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