Summability of a Jacobi Series by Lower Triangular Matrix Method

Binod Prasad Dhakal^1,

¹Tribhuvan University, Nepal

Cite this paper:
Binod Prasad Dhakal. Summability of a Jacobi Series by Lower Triangular Matrix Method. American Journal of Mathematical Analysis. 2013; 1(3):42-47. doi: 10.12691/ajma-1-3-4

Abstract

The Jacobi polynomial P_n⁽α^,β⁾(x) which is obtained from Jacobi differential equation is an orthogonal polynomial over the interval [-1, 1] with respect to weight function _(1-x)α_(1+x)β, α>-1, β>-1. Here Jacobi series has been taken and established a theorem on lower triangular matrix summability of a Jacobi series.

Keywords:
summability jacobi series triangular matrix

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References:

[1]	Beohar, B. K. and Sharma, K. G.: On Nörlund summability, Indian J. Pure Appl. Math., 11, 1475-1481. 1980.
[2]	Borwein, D.: On products of sequences, J. London Math. Soc., 33, 352-357. 1958.
[3]	Choudhary, R. S.: On Nörlund summability of Jacobi series, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 52, 644-652. 1972.
[4]	Gupta, D. P.: D.Sc. Thesis, University of Allahabad, Allahabad, 1970.
[5]	Khare, S. P. and Tripathi, S. K.: On (N,p,q) summability of Jacobi series, Indian J. Pure Appl. Math., 19( 4), 353-368. 1988.
[6]	Lal, Shyam: On the degree of approximation of conjugate of function belonging to Weighted W (Lp,ξ(t)) class by matrix summability means of conjugate series of a Fourier series, Tamkang J. Math., 31(4), 279-288. 2004.
[7]	Nörlund, N. E : Sur une application des fonctions permutables, Lund. Universities Arsskrift, 16, 1-10. 1919.
[8]	Obrechkoff, N.: Formules asymptotiques pour les polynômes de Jacobi et sur les séries suivant les memes polynômes, Ann. Univ. Sofia, Fac. Phys.-Math., 32, 39-135. 1936.
[9]	Pandey, B. N.: On the summability of Jacobi series by (N,pn) method, Indian J. Pure Appl. math., 12 (12), 1438 -1447. 1981.
[10]	Pandey, G. S. and Beohar, B. K.: On Nörlund summability of Jacobi series, Indian J. Pure Appl. Math., 9(5), 501-509. 1978
[11]	Prasad, Rajendra and Saxena, Ashok: On the Nörlund summability of Fourier-Jacobi series, Indian J. Pure Appl. Math., 10(10), 1303-1311. 1979.
[12]	Rau, H.: Über die Lebesgueschen Konstanten der Reihenentwicklungen nach Jacobischen Polynomen (German), J. f. M. , 161, 237-254. 1929.
[13]	Szegö, Gabor: Orthogonal polynomials, Amer. Math. Soc. Colloquium Publ., 23. 1959.
[14]	Chandra Satish: On double Nörlund summability of Fourier-Jacobi series, The Islamic University Journal, 15(2), 1-14, 2007.
[15]	Szili L. and Weisz F. : Uniform CesàroSummability of Jacobi-Fourier series, Acta Math. Hunger, 127(1-2), 112-138, 2010.
[16]	Thorpe, B.: Nörlund summability of Jacobi and Laguerre series, J. Reine Angew. Math., 276, 137-141. 1975.
[17]	Töeplitz, O.: Über allgemeine lineare Mittelbildungen, Prace mat. - fiz., 22, 113 -119. 1913.
[18]	Tripathi, L. M., Tripathi, V. N. and Yadav, S. J.: On Nörlund summability of Jacobi series, Math. Soc., 4, 183-193. 1988.
[19]	Zygmund, A.: Trigonometric series, Cambridge University Press, 1959.