On non-absolutely convergent integrals

Non-absolutely Convergent Integrals in Metric Spaces

A theory of non-absolutely convergent integrals in R with singularities on a regular boundary

Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in R, we are led to an integration process over quite general sets A ⊆ R with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form. Introduction. Consider an n-dimensional vector field ~v whi...

Differential Inclusions, Non-absolutely Convergent Integrals and the First Theorem of Complex Analysis

In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies C∞ regularity. As mentioned in Ahlfors [Ah 78] there have been a number of studies [Po 61], [Pl 59] proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note we will use the theory of non-absol...

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...

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 ...