scholarly journals Discrete polynomials orthogonal with respect Sobolev-type inner product associated with Chebyshev polynomials orthogonal on a uniform grid

2015 ◽  
pp. 15-20
Author(s):  
Timur Sharapudinov ◽  
◽  
Keyword(s):  
Chebyshev Polynomials ◽  
Inner Product ◽  
Uniform Grid ◽  
Sobolev Type ◽  
Discrete Polynomials

scholarly journals ON LAGUERRE–SOBOLEV TYPE ORTHOGONAL POLYNOMIALS: ZEROS AND ELECTROSTATIC INTERPRETATION

2013 ◽  
Vol 55 (1) ◽  
pp. 39-54
Author(s):  
LUIS ALEJANDRO MOLANO MOLANO
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Inner Product ◽  
Sobolev Type ◽  
Electrostatic Model ◽  
Order Linear Differential Equation ◽  
Standard Methods ◽  
Linear Differential ◽  
Electrostatic Interpretation ◽  
Image Position

AbstractWe study the sequence of monic polynomials orthogonal with respect to inner product $$\begin{eqnarray*}\langle p, q\rangle = \int \nolimits \nolimits_{0}^{\infty } p(x)q(x){e}^{- x} {x}^{\alpha } \hspace{0.167em} dx+ Mp(\zeta )q(\zeta )+ N{p}^{\prime } (\zeta ){q}^{\prime } (\zeta ),\end{eqnarray*}$$ where $\alpha \gt - 1$, $M\geq 0$, $N\geq 0$, $\zeta \lt 0$, and $p$ and $q$ are polynomials with real coefficients. We deduce some interlacing properties of their zeros and, by using standard methods, we find a second-order linear differential equation satisfied by the polynomials and discuss an electrostatic model of their zeros.

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 On the identification of the parameters of linear systems using Chebyshev polynomials of the first kind and Chebyshev polynomials orthogonal on a uniform grid

2014 ◽  
pp. 1-32
Author(s):  
Idris Sharapudinov ◽  
Murad Sultanahmedov ◽  
Tadgidin Shakh-Emirov ◽  
Magomedrasul Magomed-Kasumov ◽  
Timur Sharapudinov ◽  
...  
Keyword(s):  
Linear Systems ◽  
Chebyshev Polynomials ◽  
Uniform Grid

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.

Sign in / Sign up

Export Citation Format

Share Document