Hyper-power series and generalized real analytic functions

Uniserial modules of generalized power series

Let R be a ring, M a right R-module and (S,≤) a strictly ordered monoid. In this paper we will show that if (S,≤) is a strictly ordered monoid satisfying the condition that 0 ≤ s for all s ∈ S, then the module [[MS,≤]] of generalized power series is a uniserial right [[RS,≤]] ]]-module if and only if M is a simple right R-module and S is a chain monoid.

Generalized Analytic Functions on Generalized Domains

We define the algebra G̃(A) of Colombeau generalized functions on a subset A of the space of generalized points R̃d. If A is an open subset of R̃d, such generalized functions can be identified with pointwise maps from A into the ring of generalized numbers C̃. We study analyticity in G̃(A), where A is an open subset of C̃. In particular, if the domain is an open ball for the sharp norm on C̃, we chara...

The Real Field with Convergent Generalized Power Series

We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on [0, 1] by a series ∑ cnxn with 0 ≤ αn → ∞ and ∑ |cn|rαn < ∞ for some r > 1 is definable. This expansion is polynomially bounded.

Generalized Conjugate and Analytic Functions without Expansions.

On a Lebesgue measure space with measure element do, and total measure finite or infinite, we consider the complex-valued measurable functions f, y, so, ., each determined only a.e., and each belonging to all L,-classes simultaneously, 1<r< a. We donote (i) by F = {f} a ring of functions with complex constants and ordinary multiplication and closed under the involution: if f -= f + if2 e F then...

A Numerical Integration Method by Using Generalized Series of Functions