A Condition for $C$-Summability of Negative Order.

Summability with a Governor of Integral Order

A summability condition Qn the gradient ensuring BMO

It la well-known thaI if u E W”’(12), (2 cE .~t satisfles On E L~(fl), then u belonga te BMO(9), Ihe John-Nirenberg Space. Wc prove that Ihis ja no more true if ¡ Dii belonga te an Orlicz apace LÁ((2) whcn the N-function AQ) increases leas Ihan it. In arder te obtain u E BMO ((2), we impose a suitable uniform LA conditien for Dii

A Tauberian Theorem for (C, 1) Summability Method

In this paper, we retrieve slow oscillation of a real sequence u = (u n) out of (C, 1) summability of the generator sequence (V (0) n (Δu)) of (u n) under some additional condition. Consequently, we recover convergence or subsequential convergence of (u n) out of (C, 1) summability of (u n) under certain additional conditions that control oscillatory behavior of the sequence (u n).

On C 1 - Summability of Series

be any sequence of signs .

On Tauberian Conditions for (c, 1) Summability of Integrals

We investigate some Tauberian conditions in terms of the general control modulo of the oscillatory behavior of integer order of continuous real functions on [0,∞) for (C, 1) summability of integrals. Moreover, we obtain a Tauberian theorem for a real bounded function on [0,∞).