Open AccessLinear transformations of sequences
Publication year - 1936
Publication title -
philosophical transactions of the royal society of london. series a, mathematical and physical sciences
Language(s) - English
Resource type - Journals
eISSN - 2054-0272
pISSN - 0080-4614
DOI - 10.1098/rsta.1936.0006
Subject(s) - sequence (biology) , mathematics , combinatorics , class (philosophy) , natural number , matrix (chemical analysis) , set (abstract data type) , limit of a sequence , plane (geometry) , discrete mathematics , real number , limit (mathematics) , computer science , mathematical analysis , geometry , genetics , materials science , artificial intelligence , composite material , biology , programming language
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