Open AccessSummability of subsequences of a divergent sequence by regular matrices II
Author(s) -
Johann Boos
Publication year - 2019
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil1914509b
Subject(s) - mathematics , subsequence , proposition , sequence (biology) , conjecture , combinatorics , bounded function , matrix (chemical analysis) , discrete mathematics , pure mathematics , linguistics , mathematical analysis , philosophy , materials science , composite material , biology , genetics
C. Stuart proved in [27, Proposition 7] that the Ces?ro matrix C1 cannot sum almost every subsequence of a bounded divergent sequence. At the end of the paper he remarked ?It seems likely that this proposition could be generalized for any regular matrix, but we do not have a proof of this?. In [4, Theorem 3.1] Stuart?s conjecture is confirmed, and it is even extended to the more general case of divergent sequences. In this note we show that [4, Theorem 3.1] is a special case of Theorem 3.5.5 in [24] by proving that the set of all index sequences with positive density is of the second category. For the proof of that a decisive hint was given to the author by Harry I. Miller a few months before he passed away on 17 December 2018.