Abstract. Let ${p_{mn}}$ be a nonnegative double sequence with $p_{00}>0$ such that $(1,1)$ lies on the boundary of the convergence set of the power series (2.1), and (2.2) holds. The double sequence ${s_{mn}}$ is said to be $bJ_{p}$-limitable to $\sigma $ (in short $bJ_{p^-}\lim s_{mn} = \sigma$) if the power series (2.3) converges for $0< x,y< 1$ and $p_{s}(x,y)/p(x,y)$ is bounded and converges to $\sigma $ as $x,y \to1-$. We show (for example Satz 4.1): If $J_{p}$ is $b$-regular and ${p_{m-i, n-j}/p_{mn}}$ is a double moment sequence, with $i,j\in{-1, 0, 1}$ fixed, then $bJ_{p^-}\lim s_{mn}= \sigma $ implies $$bJ_{p^-}\lim s_{m+i, n+j} = \sigma\cdot \lim(p_{m-i, n-j}/p_{mn}).$$ This result is applied to logarithmic and generalized Abel methods.

AMS Subject Classification (1991): 40B05, 40C15

Received December 5, 1995. (Registered under 6104/2009.)