Regular Matrices with Nowhere Dense Support

Abstract

Abstract.A regular matrix has nowhere dense support in ßN\N if and only if the largest entry in the with row converges to 0.R. E. Atalla has asked [l] whether for every regular matrix A = (amn) whose rows spread (meaning limm max, | amn | = 0), the support set is nowhere dense in ßN\N. This means that every infinite subset S of N has an infinite subset P such that for every bounded sequence x:N-*R vanishing outside P the transform Ax (where Ax(m) = 2» amnx(n)) is a null sequence.Proof. Since ¿>m = max" |am"|->0, there are indices mi<mi< • • ■ such that for m>mk, bm<2~2k. Since A is regular, for each k, uniformly for i = mk+l, • ■ • , mk+i, there is j¡¡ such that 2™=;* | ay| <2~*. Any infinite set S contains an increasing sequence (tk) with tk^ijk-If x is supported by [tk\ and has sup norm M, then | 4x(tw)| <21~hM for m>mk', for in n<jk, x(n) is nonzero at most k -1 times and | amnx(n) | adds up to less than 2~kM, and for n^jk the sum is no more than 2~kM.Note that if the rows do not spread there are entries amn with m-> » such that | amn\ >e>0. By regularity, ra-►» also, and the derived set of the ra's is interior to the support set.I am indebted to Dr. Atalla for some correspondence motivating the problem. Math. Soc. 26 (1970), 437-445.

𝑅-type summability methods, Cauchy criteria, 𝑃-sets and statistical convergence

“…The support sets of multiplicative summability methods, obtained from matrices and otherwise, have been studied by a number of authors [1,12,13,18]. Atalla, in particular, used property (A) of filters to establish that the support sets of matrices are P-sets [1].…”