Note on absolutely convergent Dirichlet series

Note on Absolutely Convergent Dirichlet Series

is it true that \f(s)\ s^k>0 for cr^O implies that (/(s))-1 is also of the form (1)? In this note, an affirmative answer is supplied.3 Let P be the semigroup of positive integers under multiplication, and let h(P) be the class of all complex functions a on P, a= {an}»~i, for which ||a|| = y^°-i \an\ is finite. We obtain a commutative Banach algebra by defining (aa)n = aan for complex a, (a+b)n ...

On Absolutely Convergent Dirichlet Series

satisfying zZ\ \^A < °°, if and only if f is) is bounded away from zero in the half-plane 0-3:0. This discovery amounts to a determination of the spectrum of the Banach-algebra element associated with fis), and it thus makes available for the theory of ordinary Dirichlet series the well-known theorem of Gelfand on analytic functions of Banach-algebra elements (see §24D of [7]) and its generaliz...

Nonanalytic Functions of Absolutely Convergent Fourier Series.

On Absolutely Divergent Series

We show that in the א2-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P (ω)/fin. This complements Vojtáš’ result, that under cf(c) = p the two algebras are isomorphic [15].

Matrix Transformations from Absolutely Convergent Series to Convergent Sequences as General Weighted Mean Summability Methods

We prove the necessary and sufficient conditions for an infinity matrix to be a mapping, from absolutely convergent series to convergent sequences, which is treated as general weighted mean summability methods. The results include a classical result by Hardy and another by Moricz and Rhoades as particular cases.