Linear transformations of sequences

Given a sequence j tf8, of complex numbers, we denote by § («„) the set of all limit points of the sequence, i.e., the set of all such that lim = l p —>■ oo for some increasing sequence of positive integers If is an infinite matrix, with complex elements, and xx a sequence of complex numbers, then, formally at any rate, we can define another sequence jyK by oo y k ^ A = 1 The object of this paper is to investigate relations between $ (*A) and $ (jvK). In particular we shall be concerned with the following question. Given a class $ of matrices and a set 9ft of complex numbers, what sets £) (jv*) can be generated by sequences (#A) with *x<2ft, M < ® ? t This is a geometrical question concerning the structure of sets of points in the plane. It is therefore natural to suppose that oo lim S aKk = 1 K ► 0 0 A = 1 for every matrix of $ ; for this means only that the relationship between (*A) and $ ( y K) is unchanged by a change of origin