TY - JOUR AU - Rath, Dipty AB - TAUBERIAN THEOREMS IN THE THEORY OF ABSOLUTE RIESZ SUMMABILITY DIPTY RATH 1.1. Let Yfin be a given infinite series and {&„} a positive, steadily increasing, unbounded sequence. We write, for w ^ A lf A (w) = A °( ) = Z a , x x W n and, for fc > 0, k 1 A \w)= Z (w-A ) a = A: f (w-f)^ A (t)dt, k n n x while we suppose that for w < X A \w) = 0,k^Q. The series £#„ is said to be absolutely summable by Riesz means of type k and k k order k, or summable \R, X, k\, k ^ 0, (see [5, 6]), if w~ A (w) e BV (h, oo), h being some finite positive constant. Without loss of generality, we take 0 < h < X . 1.2. Notation. Pi = *i, Vn = K-K-i (n = 2,3,...); c = a XJn ; n n tt D(t)= Z Ac , Ac = c -c ; n n n n+1 C(t)= Z ^c; = Z A Ac ; n n where fO (H> < Aj) f (w) = (A (A