Related papers: Complex hyper-power series and generalized complex…
This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of…
Using the existence of infinite numbers $k$ in the non-Archimedean ring of Robinson-Colombeau, we define the hyperfinite Fourier transform (HFT) by considering integration extended to $[-k,k]^{n}$ instead of $(-\infty,\infty)^{n}$. In order…
Based on the full similarity in algebraic properties and differentiation rules between quaternionic (H-) holomorphic and complex (C-) holomorphic functions, we assume that there exists one holistic notion of a holomorphic function that has…
This article is the natural continuation of the paper: Mukhammadiev A.~et al Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring $\tilde{R}$ of Robinson-Colombeau is non-Archimedean, a…
In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method. In the…
The main objective of this article is a constructive generalization of the holomorphic power and Laurent series expansions in C to dimension 3 using the framework of hypercomplex function theory. For this reason, deals the first part of…
We introduce the notion of the generalized-analytical function of the poly-number variable, which is a non-trivial generalization of the notion of analytical function of the complex variable and, therefore, may turn out to be fundamental in…
We introduce and study the algebraic, analytic and lattice properties of regular homogeneous polynomials and holomorphic functions on complex Banach lattices. We show that the theory of power series with regular terms is closer to the…
The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann…
We define the algebra of Colombeau generalized functions on a subset A of the space of d-dimensional generalized points. If the domain A is open, such generalized functions can be identified with pointwise maps from A into the ring of…
Let $D$ be the open unit disc in the complex plane. We denote by $\mathbb{C}$ the set of complex numbers and consider any compact set $K$ which is disjoint from $D$ and which also has connected complement. Let $A(K)$ denote all the…
We prove a formula for the Taylor series coefficients of a zero of the sum of a complex-exponent polynomial and a base function which is a general holomorphic function with a simple zero. Such a Taylor series is more general than a Puiseux…
We study power series and analyticity in the quaternionic setting. We first consider a function f defined as the sum of a quaternionic power series centered at 0 in its domain of convergence (which is a ball B(0,R) centered at 0). At each…
A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject…
As a natural basis of the Hopf algebra of quasisymmetric functions, monomial quasisymmetric functions are formal power series defined from compositions. The same definition applies to left weak compositions, while leads to divergence for…
We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard…
The classical Kolmogorov-Gelfand theorem gives an embedding of a (compact Hausdorff) topological space X into the linear space of all linear functionals C(X)^* on the algebra of continuous functions C(X). The image is specified by algebraic…
Generalized analytic functions over generalized analytic manifolds are build from sums of convergent real power series with non-negative real exponents (and some well-ordering condition on the support). In a paper by Mart\'in-Villaverde,…
The concept of monogenic functions over real alternative $\ast$-algebras has recently been introduced to unify several classical monogenic (or regular) functions theories in hypercomplex analysis, including quaternionic, octonionic, and…
In this work, we introduce the notion of Gevrey asymptotic expansion and we show how the classical concept of a convergent power series can be generalized to include the case in which the radius of convergence is zero. This technique can be…