Surveys in Mathematics and its Applications


ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 6 (2011), 89 -- 106

A STUDY ON ALMOST MATRIX SUMMABILITY OF FOURIER-JACOBI SERIES

Hare Krishna Nigam and Ajay Sharma

Abstract. In this paper, a quite new theorem on almost summability of Fourier-Jacobi series has been established. Our theorem extends and generalizes all previously known results of this line of work.

2010 Mathematics Subject Classification: 42B05; 42B08.
Keywords:

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References

  1. B. K. Beohar and K.G. Sharma, On N\ddotorlund summability of Jacobi series, Indian J. Pure Appl. Maths., 11, no. 11, 1980, 1475-1481. MR0593587(81j:40013). Zbl 0472.40007.

  2. D. Borwein, On product of sequences, London Mathematical Society, 33, 1958, 352-357. MR0101433(21 #244). Zbl 0081.28003.

  3. Prem Chandra, Trigonometric approximation of functions in Lp norm, J. Math. Anal. Appl., 275, no. 1, 2002, 13-26. MR1941769(2003h:42003). Zbl 1011.42001.

  4. R. S. Choudhary, On N\ddotorlund summability of Jacobi series, Rend. Cl. Sci. Fis. Mat. Natur., 52, no. 8, 1970,644-652. MR0333565(48 #11890). Zbl 0274.42007.

  5. Satisha Chandra, On double N\ddotorlund summability of Fourier-Jacobi series, The Islamic University Journal,(Series of Natural studies and Engineering), 15, No.-2, 2007, 1-14.

  6. D. P. Gupta, N\ddotorlund summability of Jacobi series, D.Sc. Thesis, University of Allahabad, Allahabad, 1970.

  7. G. H. Hardy, Divergent series, Oxford University Press, first edition, 1949. MR0030620(11,25a). Zbl 0032.05801.

  8. S. P. Khare and S. K. Tripathi, On (N,p,q) summability of Jacobi series, Indian J. Pure and Appl. Math., 19, no. 4, 1988, 353-368. MR0937576(90g:40012). Zbl 0661.40004.

  9. G. G. Lorentz, A Contribution to the theory of divergent series, Acta mathematica 80, 1948, 167-190. MR0027868(10,367e). Zbl 0031.29501.

  10. N. E. N\ddotorlund, Surune application des functions permutables, Lund. Universitets Arsskrift 16, 1919, 1-10.

  11. N. Obrechkoff, Formules asymototiques pour les polynomes des Jacobi et surles series suivant les memes polynomes (Russian), Ann. Univ. Sofia Fac. Phys. Math., 32, 1936, 39-135. Zbl 0015.34701.

  12. B. N. Pandey, On the summability of Jacobi series by ( N,pn ) method, Indian J. Pure Appl. Math. 12, 1981, 1438-1447. MR0639094(83e:42031). Zbl 0495.42013.

  13. G. S. Pandey and B. K. Beohar, On N\ddotorlund summability of Jacobi series, Indian J. Pure Appl. Math. 9, no. 5, 1978, 501-509. MR0477555(57 #17074). Zbl 0389.40003.

  14. H. Rao, ÜÜber die Lebesgueschen Konstanten der Riehenewick tungennach Jcobischen polynomes, J. Reine Angew. Math. 16, 1929, 237-254.

  15. B. E. Rhoades, On the degree of approximation of functions belonging to the class Lipchitz class by Hausdorff means of its Fourier series, Tamkang Journal of Mathematics 34, 2003, no. 3, 245-247. MR2001920(2004g:41023). Zbl 1039.42001.

  16. G. Szeg\ddoto, Recent advances and open questions on the asymptotic expensions of orthogonal Polynomials, J. Soc. Indust. Appl. Math. 7, 1959. MR0116098(22 #6893).

  17. Prasad and Saxena,On the N\ddotorlund summability of Fourier-Jacobi series, Indian J. Pure Appl. Maths. 10, no. 10, 1979, 1303-1311. MR0548377(80k:42014). Zbl 0436.42019.

  18. B. Thorpe, N\ddotorlund summability of Jacobi and Laguerre series, J. Reine Angew. Math. 276, 1975, 137-141. MR0380188(52 #1088). Zbl 0303.42016.

  19. L. M. Tripathi, V. N. Tripathi and S. J. Yadav,On N\ddotorlund summability of Jacobi series, Proc. Math. Soc., B.H.U. 4, 1988, 183-193. MR1043768(91c:40017).

  20. O.Toeplitz, Uber allgemeine linear Mittelbildunger, Prace mat.-fiz., 22, 1913, 113-119. JFM 44.0281.02.


Hare Krishna Nigam Ajay Sharma
e-mail: harekrishnan@yahoo.com e-mail: ajaymathematicsanand@gmail.com

Department of Mathematics, Faculty of Engineering and Technology,
Mody Institute of Technology and Science (Deemed University),
Laxmangarh, Sikar-332311,
Rajasthan, India.



http://www.utgjiu.ro/math/sma