Kernel Integral Legendre Polynomials and Approximations
Keywords:
Legendre polynomials,Integral Legendre polynoChristoffel-Darboux formula,Fourier series expansion,Best approximation problems. , Kernel polynomialsAbstract
This paper is concerned with deriving a new system of orthogonal polynomials whose inflection points coincide with their interior roots, primitives of Legendre polynomials,they appear as solutions of linear differential equation.We study orthogonality,and extremal properties and minimization and Fourier development involving of integral Legendre polynomials. There are some important properties and certain identities and extremal properties involving both associated integral Legendre polynomials. We have used mathematical induction to establish the relation between them.We also present some results for these orthogonal polynomials by using some properties of Jacobi polynomials.General expressions are found for the kernels polynomials associated to integral Legendre polynomials.These kernel polynomials can be used to describe the approximation of continuous functions by integral Legendre polynomials.They can be used for the representation of the n-th partial sum of the Fourier series expansion of integral Legendre polynomials in the form of an integral.We conclude the paper with some results on finite Fourier series expansion by using polynomials integrals of the kernels polynomials
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References
M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions, 10th Edition, Dover, New York, 1972.
2 : F. Alberto Grunbaum, Random walks and orthogonal polynomials: some challenges, Probability, Geometry and Integrable Systems MSRI Publications,2007,Volume 55.
3 : R. Askey, Orthogonal Polynomials and Special Functions, Regional Conf. Ser.Appl. Math. 21, SIAM, Philadelphia,1975.
4 : S. Belmehdi and S.Lewanowicz and A. Ronveaux, Linearization of the product of orthogonal polynomials of a discrete variable .Applicationes Mathematicae.
24,4 (1997), pp. 445--455.
5 : R. Belinsky, Integrals of Legendre polynomials and solution of some partial differential equations, Journal of Applied Analysis, Vol. 6, No. 2 (2000), pp. 259--282.
6 : J. P. Boyd, Chebyshev and Fourier spectral methods, (Dover, 2nd edn., Mineola, 2001).
7 : T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach,New York, 1978.
8 : K. Dilcher, K. B. Stolarsky, Resultants and Discriminants of Chebyshev and related polynomials, Transactions of the Amer. Math. Soc. 357 (2004), 965-981.
9 : M. Foupouagnigni, On difference and differential equations for modifications of classical orthogonal polynomials. Habilitation Thesis, Department of Mathematics and Computer Science. University of Kassel, Kassel, January 2006.
10 : Reza Dehghan, Chelyshkov polynomials approximation for solving a class of linear and nonlinear equations arising in astrophysics,Mathematics and Computational Sciences,Volume 5, Issue 3, 2024, Pages 19-37, https://doi.org/10.30511/mcs.2024.2020655.1151
11 : Mahsa Zaboli ,Haleh Tajadodi,A spectral collocation method for solving stochastic fractional integrodifferential equation,Mathematics and Computational Sciences,Vol.6(2) 2025, Pages 1-31,
12 : Meisam Noei Khorshidi, Mohammad Arab Firoozjaee,Legendre Ritz-Least squares method for the numerical solution of delay differential equations of the multi-pantograph type Mathematics and Computational Sciences, Vol. 5(1) 2024, Pages 20-29, https://doi.org/10.30511/mcs.2024.2007870.1133
13 : Y Esmaeelzade Aghdam,B Farnam,A numerical process of the mobile-immobile advection-dispersion model arising in solute transport Mathematics and Computational Sciences, Vol 3(3) 2022, Pages 1-10 https://doi.org/10.30511/mcs.2022.554606.1072
14 : Jemal Emina Gishe, A finite family of q-orthogonal polynomials and resultants of Chebyshev polynomials. Graduate School Theses and Dissertations. University of South Florida
15 : K. H. Kwon, L. L. Little John, Table of Integrals, Series and Products. Academic Press (1980).
16 : F. Marcellan, J. Petrolinho, Orthogonal polynomials and quadratic transformations, Portugaliae Mathematica, Vol.56 Fasc.1(1999)
17 : J. C. Mason, D. C. Handscomb, Chebyshev polynomials, (Chapman&Hall, London, 2003)
18 : G. V. Milovanović, A. S. Cvestković, M. M. Matejić, Remark on orthogonal polynomials induced by the modified Chebychev measure of the second kind. Facta Universitatis ( Nis) Ser. Math.Inform, 21, 2006, pp. 13-21.
19 : H.Pijeira.Cabrera and Y.José Bello Cruz and W.Urbina ,On Polar Legendre polynomials,Rocky Mountain Journal of Mathematics,RMJ (2008),vol 38,1-10.
20 : A. Rehouma and M. J. Atia, Study of an example of Markov chains involving Chebyshev polynomials. Integral transforms and special functions. Received 22January 2022. Accepted 1 July 2022.
21 : J. Shen, T. Tang, L.-L. Wang, Spectral methods: algorithms, analysis and applications, (Springer,2011).
22 : G. Szego ,Orthogonal Polynomials, American Mathematical Society , New York, 1939.
23 : O.Farrell, B.Ross, Solved Problems, The Macmollan Company, New York, 1963.
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