STUDY ON ARITHMETICAL SUMMABILITY AND ITS APPLICATIONS

Abstract

Studies on the summability methods as a class of transformations of complex sequences newlineby complex infinite matrices was introduced by O. Toeplitz. This subject newlinewas further studied by many other mathematicians like T. Kojima, H. Steinhaus, newlineI. Schur, S. Mazur, W. Orlicz, K. Knopp, T. Pati, R.P. Agnew, A. Brudnow, R.G. newlineCooke, V.G. Iyer, G.M. Peterson and many others. With the emergence of functional newlineanalysis, sequence spaces were studied with greater insight and motivation. newlineAn earliest application of functional analysis to summability was made by S. Banach, newlineH. Hahn, S. Mazur, G. K¨othe and O. Topelitz. Later on it was further newlinestudied by G.G. Lorentz, K. Zeller, A. Wilansky, D.C. Russell, W.L.C. Sargent, newlineB.E. Rhoades, I.J. Maddox and many others. newlineTo extend the concept of summability, W.H.Ruckle [61] introduced the notions newlineof Arithmetic Summability and Arithmetic Convergence. In his paper he defined a newlinetheory of summability using the notation newlineP newlinek|m f(k) which is the finite sum of all newlinethe numbers f(k) as k ranges over the integers that divide m including 1 and m. newline