Absolutely convergent Lidstone series, II

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.

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 ...

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.

Concerning Hausdorff Matrices and Absolutely Convergent Sequences1

for£}=0, 1,2,.... This paper is concerned with the determination of those generalized Hausdorff matrices which sum all absolutely convergent sequences. Theorem 2 gives a sufficiency condition stated in terms of the behavior of the points of continuity of the graph of g and Theorems 3 and 4 provide examples which serve to further describe functions which generate such matrices. Theorem 5 gives a...