Related papers: Taylor expansions over generalised power series
The theory of Touchard polynomials is generalized using a method based on the definition of exponential operators, which extend the notion of the shift operator. The proposed technique, along with the use of the relevant operational…
Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} \omega_n \frac{z^n}{n!} \] with unimodular coefficients $|\omega_n|=1$. We show that either the counting function $n_F(r)$ of zeroes…
Let $T$ be the theory of an o-minimal field and $T_0$ a common reduct of $T$ and $T_{an}$. I adapt Mourgues' and Ressayre's constructions to deduce structure results for $T_0$-reducts of $T$-$\lambda$-spherical completion of models of…
In transferring some results from universal Taylor series to the case of Pad\'e approximants we obtain stronger results, such as, universal approximation on compact sets of arbitrary connectivity and generic results on planar domains of any…
Power series are introduced that are simultaneously convergent for all real and p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and…
Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions…
Generalizations of classical theta functions are proposed that include any even number of analytic parameters for which conditions of quasi-periodicity are fulfilled and that are representations of extended Heisenberg group. Differential…
Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we…
A preparation theorem for compositions of restricted log-exp-analytic functions and power functions of the form $$h: \mathbb{R} \to \mathbb{R}, x \mapsto \left\{\begin{array}{ll} x^r, & x > 0, \\ 0, & \textnormal{ else, }…
Extending the work of Freese, we further develop the theory of generalized trigonometric functions. In particular, we study to what extent the notion of polar form for the complex numbers may be generalized to arbitrary associative…
The differential $\lambda$-calculus studies how the quantitative aspects of programs correspond to differentiation and to Taylor expansion inside models of linear logic. Recent work has generalized the axioms of Taylor expansion so they…
Pointwise-supported generalized wavelets are introduced, based on Dirac, doublet and further derivatives of delta. A generalized biorthogonal analysis leads to standard Taylor series and new Dual-Taylor series that may be interpreted as…
A class theorem is presented and proved: the complex Fourier transforms of a certain class of exponential functions have all their zeros on the real line. A class of basis functions is first considered, and the class is then extended via…
We work with semi-algebraic functions on arbitrary real closed fields. We generalize the notion of critical values and prove a Sard type theorem in our framework.
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
In this note, we study substructures of generalised power series fields induced by families of well-ordered subsets of the group of exponents. We characterise the set-theoretic and algebraic properties of the induced substructures in terms…
We prove simultaneous Universal Approximation of a certain type of Pade Approximants and of Taylor series with the same indexes. This is a generic phenomenon in the space of holomorphic functions in any simply connected domain, as well as…
Tensor fields depending on other tensor fields are considered. The concept of extended tensor fields is introduced and the theory of differentiation for such fields is developed.
A class of overlooked trigonometric-like functions is explored in this article, along with the relevant applications in applications. We show indeed that Taylor series, resembling that of an ordinary cosine, are representative of wider…
It is well known that any power series over a finite field represents a rational function if and only if its sequence of coefficients is ultimately periodic. The famous Christol's Theorem states that a power series over a finite field is…