On the set of points with absolutely convergent trigonometric series

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 ...

Nonanalytic Functions of Absolutely Convergent Fourier Series.

Uniqueness for spherically convergent multiple trigonometric series

In 1870 Cantor proved that representation of a function of one variable by a trigonometric series can be done in only one way. In 1996 Bourgain proved the same thing for spherical convergence and multiple trigonometric series. His proof involves injecting a lot of new ideas into the theory of uniqueness. We give here an exposition of Bourgain’s proof, specialized to the case of dimension 2. Our...

On the Set of Points of Convergence of a Lacunary Trigonometric Series and the Equidistribution Properties of Related Sequences

It is well known from the classical theory of trigonometric series that the series (1) cannot converge except possibly for values of x in a set of zero Lebesgue measure . Our object is to discover how `thin' this set is for various types of sequence {nk } . The first result of this kind is due to P . Turan who proved, in 1941, that the series (1) converges absolutely in a set of positive logari...