Absztrakt: | The author has extended the result of {\it G. H. Hardy} [A theorem
concerning summable series, Cambr. Phil. Soc. Proc. 20, 304-307
(1921; JFM 48.1190.01)] for summability $(C,1)$ of a series of
complex numbers $\sum^\infty_{n= 0}a_n$ by the convergence of
another series $\sum^\infty_{n= 0} b_n$ where $b_n= \sum^\infty_{k=n}
a_k/k+1$, $k= 0,1,2,\dots$ to series whose terms are elements
of a Banach space and to integrals of locally integral functions
over $\bbfR_+= [0,\infty]$. The analogue of the result $s_n/n\to
0$ as $n\to\infty$ is not true for integrals. Also Tauberian
conditions for integrals are established from which convergence
follows from $(C,1)$ summability of integrals.}
\RV{I.L.Sukla (Orissa) |