ON ABSOLUTE ALMOST MATRIX SUMMABILITY OF ORTHOGONAL SERIES

In this paper we present some results on absolute almost matrixsummability of an orthogonal series. Precisely, some sufficient conditions under which an orthogonal series will be absolute almost matrix summable areobtained. The most important corollaries of the main results also are deduced

Keywords:

Orthogonal series, almost matrix summability absolute summability,

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Das, G. and Ray, B. K., Lack of Tauberian theorem for absolute almost convergence. Anal. Math. 35 (2009), 37-49.
Das, G. and Kuttner, B., S pace of absolute almost convergence. Indian J. Math. 28 (1986), 241-257.
Das, G., Kuttner, B. and Nanda, S., S ome sequence spaces and absolute almost convergence. Trans. Amer. Math. Soc. 283 (1984), no. 2, 729-739.
Flett, T. M., O n an extension of absolute summability and some theorems of Littlewood and Paley. Proc. London Math. Soc. 7 (1957), 113-141.
Lorentz, G. G., A contribution to the theory of divergent series. Acta Math. 80 (1948), 167-190.
Mazhar, S. M. and Siddiqi, A. H., O n olmost summability of a trigonometric sequence. Acta Math. Acad. Sci. Hungar. 20 (1969), 21-24. [7] Leindler, L., ¨Uber die absolute summierbarkeit der orthogonalreihen. (German) Acta Sci. Math. (Szeged) 22 (1961), 243-268.
Leindler, L., O n the absolute Riesz summability of orthogonal series. Acta Sci. Math. (Szeged) 46 (1983), no. 1-4, 203-209.
Leindler, L. and Tandori, K., O n absolute summability of orthogonal series. Acta Sci. Math. (Szeged) 50 (1986), no. 1-2, 99-04.
Leindler, L., O n the newly generalized absolute Riesz summability of orthogonal series. Anal. Math. 21 (1995), no. 4, 285-297.
Tandori, K. ¨Uber die orthogonalen Funktionen IX (Absolute Summation). Acta Sci. Math. (Szeged) 21 (1960), 292-299.
Okuyama, Y. and Tsuchikura, T., O n the absolute Riesz summability of orthogonal series. Anal. Math. 7 (1981), 199-208.
Okuyama, Y., O n the absolute N¨orlund summability of orthogonal series. Proc. Japan Acad. 54 (1978), 113-118.
Okuyama, Y., O n the absolute generalized N¨orlund summability of orthogonal series. Tamkang J. Math. 33 (2002), no. 2, 161-165. [15] Okuyama, Y., O n the absolute Nrlund summability of orthogonal series. Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), no. 5, 113-118. [16] Okuyama, Y., O n the absolute Riesz summability of orthogonal series. Tamkang J. Math. 19 (1988), no. 3, 75-89.
Szalay, I., O n generalized absolute Ces`aro summability of orthogonal series. Acta Sci. Math. (Szeged) 32 (1971), 51-57.
Billard, P., S ur la sommabilit´e absolue des s´eries de functions orthogonales. (French) Bull. Sci. Math. 85 (1961), no. 2, 29-33.
Grepachevskaya, L. V., Absolute summability of orthogonal series. (Russian) Mat. Sb. (N.S.) 65 (1964), 370-389.
Spevakov, V. N. and Kudrjavcev, B. A., Absolute summability of orthogonal series by the Euler method. (Russian) Math. Notes 21 (1977), no. 1, 51-56.
Lal, S., O n the approximation of function belonging to weighted (Lp, ξ(t)) class by almost matrix summability method of its Fourier series. Tamkang J. Math. 35 (2004), no. 1, 67–76. [22] Qureshi, K., O n the degree of approximation of a periodic function f by almost N¨orlund means. Tamkang J. Math. 12 (1981), no. 1, 35-38.
Ulyanov, P. L., S olved and unsolved problem in the theory of trigonometric and orthogonal series. Uspehi Math. Nauk. 19 (1964), 3-69. [24] Krasniqi, Xh. Z., A note on |N, p, q|k(1 ≤ k ≤ 2) summability of orthogonal series. Note Mat. 30 (2010), 135-139.
Krasniqi, Xh. Z., O n absolute weighted mean summability of orthogonal series. Sel¸cuk J. Appl. Math. Appl. 12 (2011), no. 2 63-70.
Krasniqi, Xh. Z., O n absolutely almost convergency of higher order of orthogonal series. Int. J. Open Problems Comput. Sci. Math. 4 (2011), no. 1, 44-51.
Krasniqi, Xh. Z., O n |A, δ|k−summability of orthogonal series. Math. Bohem. 137 (2012), no. 1, 17-25.
Krasniqi, Xh. Z., O n absolute almost generalized N¨orlund summability of orthogonal series. Kyungpook Math. J. 52 (2012), 279-290.
Tanaka, M., O n generalized N¨orlund methods of summability. Bull. Austral. Math. Soc. 19, (1978), 381-402.
Hardy, G. H., D ivergent Series. First edition, Oxford University Press, 1949.
Alexits, G., C onvergence problems of orthogonal series. Translated from the German by I. F¨older. International Series of Monographs in Pure and Applied Mathematics, Vol. 20, Pergamon Press, New York-Oxford-Paris 1961, 350 pp.
Okuyama, Y., Absolute summability of Fourier series and orthogonal series. Lecture Notes in Mathematics, 1067. Springer-Verlag, Berlin, 1984, 118 pp.
Department of Mathematics and Computer Sciences, University of Prishtina, Avenue ”Mother Theresa ” 5, Prishtin¨e 10000, KOSOV ¨E
E-mail address: xhevat.krasniqi@uni-pr.edu, xhevat-z-krasniqi@hotmail.com