Mathema Tics: C. R. Adams on Non-factorable Transformations of Double Sequences'
نویسنده
چکیده
defined by a four-dimensional "triangular" matrix jamnkjjj (m, n, k, I = 1, 2, 3, ...; amnkl = O for k > m or I > n or both) of real or complex numbers. We showed (Theorem 2 of I) that if (A) is regular for bounded sequences, it is in general regular also for a large class of unbounded sequences. Special interest attaches to the type of transformation (A) for which we have amnklz = amk anl (m, n, k, I = 1, 2, 3, .. .), a mk and a"', being, respectively, the general elements of two triangular matrices; this type we call factorable, and we write A = A 'GA ". For a factorable transformation to be regular for bounded sequences it is necessary and sufficient3 that each factor matrix be a pure C-matrix in the sense of Hausdorff4 (i.e., a matrix defining a simple sequence transformation which is convergence-preservng and regular for null sequences) and together they satisfy the condition l'tl" = 1, where
منابع مشابه
MATHEMA TICS: A. D. MICHAL THE GROUP MANIFOLD OF FINITE CONTINUOUS POINT AND FUNCTIONAL TRANSFORMATION GROUPS By ARIsTomII
has recently been studied by Cartan' with the aid of ingenious geometric methods peculiar to his way of looking at an affinely connected manifold.2 From the analytic and invariant-theoretic standpoint, however, Cartan's geometric methods are by no means the last word on the subject. Furthermore, the proof of many of his theorems bring into play the variables xl, x2, .. ., x". It is the purpose ...
متن کاملFunctions preserving slowly oscillating double sequences
A double sequence x = {xk,l} of points in R is slowly oscillating if for any given ε > 0, there exist α = α(ε) > 0, δ = δ(ε) > 0, and N = N(ε) such that |xk,l−xs,t| < ε whenever k, l ≥ N(ε) and k ≤ s ≤ (1 + α)k, l ≤ t ≤ (1 + δ)l. We study continuity type properties of factorable double functions defined on a double subset A × A of R into R, and obtain interesting results related to uniform cont...
متن کامل