Related papers: Generalized universal series

In this paper we present a generalization of Faulhaber's formula to sums of arbitrary complex powers $m\in\mathbb{C}$. These summation formulas for sums of the form $\sum_{k=1}^{\lfloor x\rfloor}k^{m}$ and $\sum_{k=1}^{n}k^{m}$, where…

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 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…

We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We…

We provide a new foundational approach to the generalization of terms up to equational theories. We interpret generalization problems in a universal-algebraic setting making a key use of projective and exact algebras in the variety…

Let $(a_n)_{n\geq 0}$ be an arbitrary sequence and $(a_{\lfloor n/k \rfloor})_{n\geq 0}$ its dual floor sequence. We study infinite series and finite generalized binomial sums involving $(a_{\lfloor n/k \rfloor})_{n\geq 0}$. As applications…

We extend calculus from smooth manifolds to topological manifolds making use of a theory of generalized functions developed for this aim. Actually such extension fits into a boarder context: the universal construction of a site containing…

Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…

Using a recent Mergelyan type theorem, we show the existence of universal Taylor series on products of planar simply connected domains Oi that extend continuously on the product of the union of Oi with Si , where Si are subsets of the…

The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the…

We prove a result about the non-existence of certain sums-of-squares formulas over a field. This generalizes an old theorem which used topological K-theory to obtain obstruction conditions when the field is the real numbers. Our result…

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…

Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin…

We prove the existence of holomorphic functions $f$ defined on any open convex subset ${\rm \Omega}\subset {{\mathbb C}}^n$, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact…

We show generic existence of power series a with complex coefficients a_n, such that the sequence of partial sums of a new power series where its coefficients b_n are functions of a_0, a_1, ..., a_n approximate every polynomial uniformly on…

We discuss algebraic universality in the sense of P. Vogel for the simplest refined quantity, the Macdonald dimensions. The main known source of universal quantities is given by Chern-Simons theory. Refinement of Chern-Simons theory means…

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…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

We extend the construction of [19] by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby…

We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is…