Nonabsolutely convergent series

Nonabsolutely convergent Poisson integrals

If a function f has finite Henstock integral on the boundary of the unit disk of R 2 then its Poisson integral exists for |z| < 1 and is o((1 − |z|) −1) as |z| → 1 −. It is shown that this is the best possible uniform pointwise estimate. For an L 1 measure the best estimate is O((1 − |z|) −1). In this paper we consider estimates of Poisson integrals on the unit circle with respect to Alexiewicz...

The Existence of Nonabsolutely Convergent Fourier Series on Compact Groups

1. J. W. S. Cassels, An introduction to diophantine approximation, Cambridge University Press, 1959, Chapter IV, §5. 2. J. F. Koksma, Diophantische Approximationen, Ergebnisse der Mathematik, vol. IV, Berlin, Springer, 1937, Chapter VIII, §3. 3. G. M. Petersen, Almost convergence and uniformly distributed sequences, Quart. J. Math. vol. 7 (1956) pp. 188-191. 4. H. Weyl, Uber die Gleichverteilun...

A Nonabsolutely Convergent Integral Defined by Partitions of the Unity

In [5], a type of integral defined by partitions of the unity (PU-integral) is defined on an abstract compact metric measure space and it is proved that a PUintegrable function is μ−integrable and that the μ−integral is equivalent to the PU-integral. Moreover an example of a non Euclidean space on which is defined this type of integral is given. The PU-integral is obtained by approssimations of...

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...

On Cantor's Series with Convergent