J ournal of Classical Analysis Volume 14, Number 1 (2019), 49–55 doi:10.7153/jca-2019-14-05 FOUR DIMENSIONAL LOGARITHMIC TRANSFORMATION INTO L u FATIH NURAY AND NIMET AKIN Abstract. Let t =(t m ) and t =( t n ) be two null sequences in the interval (0, 1) and define the four dimensional logarithmic matrix L t , t =(a t , t mnkl ) by a t , t mnkl = 1 log(1 - t m ) log(1 - t n ) 1 (k + 1)(l + 1) t k+1 m ( t n ) l +1 . The matrix L t , t determines a sequence -to-sequence variant of classicial logarithmic summabil- ity method. The aim of this paper is to study these transformations to be L u - L u summability methods. Mathematics subject classification (2010): 40B05,40C05. Keywords and phrases: Tauberian condition, logarithmic summability, four dimensional summability method, double sequences, Pringsheim limit. REFERENCES [1] R. P. AGNEW,Inclusion relations among methods of summability compounded form given matrix methods, Ark. Mat. 2, (1952), 361–374. [2] F. BAS ¸ AR, Summability Theory and Its Applications, Bentham Science Publishers, e-books, Mono- graphs, ˙ Istanbul, (2012). [3] J. A. FRIDY, Absolute summability matrices that are stronger than the identity mapping, Proc. Amer. Math. Soc. 47, (1995), 112–118. [4] J. A. FRIDY AND K. L. ROBERT, Some Tauberian theorems for Euler and Borel tummability, Intnat. J. Math. & Math. Sci.3,4 (1980), 731–738. [5] J. A. FRIDY, Abel transformations into l 1 , Canad. Math. Bull, 25, (1982), 421–427. [6] H. J. HAMILTON, Transformations of multiple sequences, Duke Math. J., 2 (1936), 29–60. [7] G. H. HARDY, Divergent series, Oxford, (1949). [8] G. H. HARDY AND J. E. LITTLEWOOD, Theorems concerning the summability of series by Borel’s exponential methods, Rend. Circ. Mat. Palermo, 41, (1916), 36–53. [9] G. H. HARDY AND J. E. LITTLEWOOD, On the Tauberian theorem for Borel summability, J. London Math. Soc., 18, (1943), 194–200. [10] M. I. KADETS, On absolute, perfect, and unconditional convergences of double series in Banach spaces, Ukrainian Math. J., 49, 8 (1997), 1158–1168. [11] M. LEMMA, Logarithmic transformations into l 1 , Rocky Mountain J. Math. ,28, 1 (1998), 253–266. [12] M. MURSALEEN AND S.A. MOHIUDDINE, Convergence Methods for Double sequences and Appli- cations, Springer Briefs In Mathematics, 2013. [13] R. F. PATTERSON, A theorem on entire four dimensional summability methods, Appl. Math. Comput., 219, (2013), 7777–7782. [14] R. F. PATTERSON, Four dimensional matrix characterization of absolute summability, Soochow J. Math., 30, 1 (2004), 21–26. [15] R. F. PATTERSON, Analogues of some fundamental theorems of summability theory, Internat. J. Math. & Math. Sci., 23, 1 (2000), 1–9. [16] A. PRINGSHEIM, Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann.,53, (1900), 289–32. [17] G. M. ROBISON,Divergent double sequences and series, Trans. Amer. Math. Soc., 28, (1926), 50–73. c  , Zagreb Paper JCA-14-05