Effects of linear transformations on the divergence of bounded sequences and functions

where { xi } is a sequence of complex elements and the Kn,i are complex numbers, has been widely studied, and the conditions which must be fulfilled by the Kn,i in order that the property of convergence of the sequence may remain invariant were given by Schur [I ].t In recent studies by Hurwitz [2, 3] and Knopp [4] modes of measuring the divergence of bounded sequences were given, and the conditions on the Kn,i were found under which the divergence of the sequence { yn } is no greater than that of {Xn}. In this paper the effects of the transformations will be investigated with fewer restrictions on the Kn,i than those imposed by earlier writers. The problem will be approached by means of the new concept of the limit circle defined as follows: The limit circle of a bounded sequence of complex elements is the (unique) circle of least radius which contains within or on its boundary the limit points of the sequence. The limit circle of a bounded function F(y) of the complex variable y as y->t (finite or infinite) is analogously defined in terms of the limit points of F(y) as y->t; this concept will be used in the study of transformations of sequences and functions into functions. 2. Sequence to function transformations. Instead of the transformation mentioned in the introduction we shall study the following more general transformation S. Let T be a set of points in the complex plane having a limit point to (finite or infinite) not belonging to T. We shall speak of a point t in T as being sufficiently advanced if for some a >0, jt toI < when to is finite, or j i/tI < 5 when to is infinite. Then let Ki(t) be a set of complex numbers defined for i= 1, 2, ... , and each t in T, and such that