Algebraic genericity and the differentiability of the convolution
type: Item
instance of: scholarly article
title: Algebraic genericity and the differentiability of the convolution
zbMATH Open document ID: 1417.42009
author: P. Jiménez-Rodríguez
author: G. A. Muñoz-Fernández
author: Eva Sáez-Maestro
author: J. B. Seoane-Sepúlveda
published in: Journal of Approximation Theory
publication date: 2019-03-27T00:00:00Z
review text: The convolution of the $2$-periodic functions $f, g \in L^1[-1, 1]$ is defined as the function $(f * g)(x) = \int_{-1}^1 f(s)g(x-s)ds$. It is well known that, for instance, if $f \in C^k[-1,1]$ then $(f*g) \in C^k[-1,1]$ without any kind of requirement on $g$. However, there are examples of two differentiable functions whose convolution is not differentiable. \par In this interesting paper, the authors focus their efforts on lineability, algebrability, and coneability within this context, that is, they look for the existence of large algebras of the biggest possible dimension formed by differentiable functions (with the null vector not included) whose convolution is not differentiable. More specifically, they prove that there exist two algebras $V$ and $W$ (both of dimension $\mathfrak{c}$) in $D[-1, 1]$ ($2$-periodic differentiable functions) so that, if $f \in V \setminus \{0\}$ and $g \in W \setminus \{0\}$, then $(f * g)$ is not differentiable at $0$. \par In particular, they show that every function in $C[-1, 1]$ (continuous $2$-periodic functions) can be approximated by vectors taken from a linear space consisting of differentiable functions that (except for 0) may be paired with another differentiable function so that the resulting convolution is not differentiable at $0$.
reviewed by: Jamilson Ramos Campos
Mathematics Subject Classification ID: 42A85
Mathematics Subject Classification ID: 58B10
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 7043341
zbMATH Keywords: convolution
zbMATH Keywords: nondifferentiable function
zbMATH Keywords: lineability
zbMATH Keywords: spaceability
zbMATH Keywords: algebrability
MaRDI profile type: Publication
full work available at URL: j.jat.2019.01.002
OpenAlex ID: W2912012841
cites work: Q3531317
cites work: Lineability
cites work: Lineability and spaceability of sets of functions on $\mathbb {R}$
cites work: Lineability criteria, with applications
cites work: Linear subsets of nonlinear sets in topological vector spaces
cites work: Banach-Steinhaus theory revisited: lineability and spaceability
cites work: Spaceability and strong divergence of the Shannon sampling series and applications
cites work: Lineability of non-differentiable Pettis primitives
cites work: Self-similar functions, fractals and algebraic genericity
cites work: Basic sequences and spaceability in \(\ell_p\) spaces
cites work: Generalized universal series
cites work: Cardinal coefficients related to surjectivity, Darboux, and Sierpiński-Zygmund maps
cites work: On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity
cites work: Algebras of measurable extendable functions of maximal cardinality
cites work: Differentiability versus continuity: Restriction and extension theorems and monstrous examples
cites work: Lineability within probability theory settings
cites work: Some results and open questions on spaceability in function spaces
cites work: Q3138785
cites work: Q5577487
cites work: Sur le produit de composition de deux fonctions continues
cites work: Convolution functions that are nowhere differentiable
cites work: The convolution of two differentiable functions on the circle need not be differentiable
cites work: Q5556050
cites work: Locally recurrent functions, density topologies and algebraic genericity
cites work: A note on latticeability and algebrability
cites work: Bounded linear non-absolutely summing operators
Wikidata QID: Q128549762
DOI: 10.1016/J.JAT.2019.01.002
Recommended article: Q4729940
Recommended article: Generalized derivations on some convolution algebras
Recommended article: Generalized differentiation and characterizations for differentiability of infimal convolutions
Recommended article: Generalized convolution behaviors and topological algebra
Recommended article: On algebras generated by convolutions and discontinuous functions
Recommended article: Generalization of convolution and incidence algebras
Recommended article: Q3477492
Recommended article: Q4381312
Recommended article: DERIVATIONS ON CONVOLUTION ALGEBRAS
Recommended article: Q4872748
description: scientific article; zbMATH DE number 7043341