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J. M. Almira

Publications and source records attributed to J. M. Almira.

At least 19 recordsLinked to original sources
arXiv2026

Polynomial maps on the monoid of words

We briefly visit the theory of polynomial and semipolynomial maps defined on an arbitrary monoid, with range a commutative group. Then we characterize the space $\mathcal{P}(S,\mathbb{C})$ of polynomial maps $f:S\to \mathbb{C}$, where $S=\mathcal{A}^*$ is the monoid of words based on an arbitrary alphabet $\mathcal{A}$ under concatenation, and we use this characterization to prove that if there exists a monoid $S\not\in\mathcal{CS}$ such that $\mathcal{SP}(S,\mathbb{C})\neq \mathcal{P}(S,\mathbb{C})$, then also $\mathcal{SP}(\mathcal{A}^*,\mathbb{C})\neq\mathcal{P}(\mathcal{A}^*,\mathbb{C})$ for a certain alphabet $\mathcal{A}$. We propose as an open problem to prove or disprove that $\mathcal{SP}(\mathcal{A}^*,\mathbb{C})=\mathcal{P}(\mathcal{A}^*,\mathbb{C})$ for arbitrary alphabets $\mathcal{A}$. Our results are motivated by previous work of Shulman.

math.GR↗
arXiv2024

Using the Baire Category Theorem to Explore Lions Problem for Quasi-Banach Spaces

Many results for Banach spaces also hold for quasi-Banach spaces. One important such example is results depending on the Baire Category Theorem (BCT). We use the BCT to explore Lions problem for a quasi-Banach couple $(A_0, A_1)$. Lions problem, posed in 1960's, is to prove that different parameters $(θ,p)$ produce different interpolation spaces $(A_0, A_1)_{θ, p}$. We first establish conditions on $A_0$ and $A_1$ so that interpolation spaces of this couple are strictly intermediate spaces between $A_0+A_1$ and $A_0\cap A_1$. This result, together with a reiteration theorem, gives a partial solution to Lions problem for quasi-Banach couples. We then apply our interpolation result to (partially) answer a question posed by Pietsch. More precisely, we show that if $p\neq p^*$ the operator ideals $\mathcal{L}^{(a)}_{p,q}(X,Y)$, $\mathcal{L}^{(a)}_{p^*,q^*}(X,Y)$ generated by approximation numbers are distinct. Moreover, for any fixed $p$, either all operator ideals $\mathcal{L}^{(a)}_{p,q}(X,Y)$ collapse into a unique space or they are pairwise distinct. We cite counterexamples which show that using interpolation spaces is not appropriate to solve Pietsch's problem for operator ideals based on general $s$-numbers. However, the BCT can be used to prove a lethargy result for arbitrary $s$-numbers which guarantees that, under very minimal conditions on $X,Y$, the space $\mathcal{L}^{(s)}_{p,q}(X,Y)$ is strictly embedded into $\mathcal{L}^{\mathcal{A}}(X,Y)$. The paper is dedicated to the memory of Prof. A. Pietsch, who passed away recently.

math.FA↗
arXiv2022

Aichinger equation on commutative semigroups

We consider Aichinger's equation $$f(x_1+\cdots+x_{m+1})=\sum_{i=1}^{m+1}g_i(x_1,x_2,\cdots, \widehat{x_i},\cdots, x_{m+1})$$ for functions defined on commutative semigroups which take values on commutative groups. The solutions of this equation are, under very mild hypotheses, generalized polynomials. We use the canonical form of generalized polynomials to prove that compositions and products of generalized polynomials are again generalized polynomials and that the bounds for the degrees are, in this new context, the natural ones. In some cases, we also show that a polynomial function defined on a semigroup can uniquely be extended to a polynomial function defined on a larger group. For example, if $f$ solves Aichinger's equation under the additional restriction that $x_1,\cdots,x_{m+1}\in \mathbb{R}_+^p$, then there exists a unique polynomial function $F$ defined on $\mathbb{R}^p$ such that $F_{|\mathbb{R}_+^p}=f$. In particular, if $f$ is also bounded on a set $A\subseteq \mathbb{R}_+^p$ with positive Lebesgue measure then its unique polynomial extension $F$ is an ordinary polynomial of $p$ variables with total degree $\leq m$, and the functions $g_i$ are also restrictions to $\mathbb{R}_+^{pm}$ of ordinary polynomials of total degree $\leq m$ defined on $\mathbb{R}^{pm}$.

math.AC↗
arXiv2022

Using Aichinger's equation to characterize polynomial functions

Aichinger's equation is used to give simple proofs of several well-known characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger's equation characterizes polynomial functions to solve, for arbitrary commutative groups, Ghurye-Olkin's functional equation, Wilson's functional equation, the Kakutani-Nagumo-Walsh functional equation, and a general version of Fréchet's unmixed functional equation.

math.CA↗
arXiv2020

Negative results for approximation using single layer and multilayer feedforward neural networks

We prove a negative result for the approximation of functions defined on compact subsets of $\mathbb{R}^d$ (where $d \geq 2$) using feedforward neural networks with one hidden layer and arbitrary continuous activation function. In a nutshell, this result claims the existence of target functions that are as difficult to approximate using these neural networks as one may want. We also demonstrate an analogous result (for general $d \in \mathbb{N}$) for neural networks with an \emph{arbitrary} number of hidden layers, for activation functions that are either rational functions or continuous splines with finitely many pieces.

cs.LG↗
arXiv2020

On Popoviciu-Ionescu functional equation in one and several variables

We study a functional equation first proposed by T. Popoviciu in 1955. In the one-dimensional case, it was solved for the easiest case by Ionescu in 1956 and, for the general case, by Ghiorcoiasiu and Roscau and Radó in 1962. We present a solution to the equation both for the one-dimensional and the higher-dimensional cases, which is based on a generalization of Radó's theorem to distributions in a higher dimensional setting. For functions of a single variable, our solution is different from the existing ones and, for functions of several variables, our solution is, as far as we know, the first one that exists. The one-dimensional part already appeared in a previous paper by the author. Indeed, this paper is just a re-statement of the one-dimensional part of the paper Arxiv:1604.00616, with the addition of the arguments devoted to solve the equation in dimensions higher than one.

math.CA↗
arXiv2019

A new proof of a classical result on the topology of orientable connected and compact surfaces by means of the Bochner technique

As an application of the Bochner formula, we prove that if a $2$-dimensional Riemannian manifold admits a non-trivial smooth tangent vector field $X$ then its Gauss curvature is the divergence of a tangent vector field, constructed from $X$, defined on the open subset out the zeroes of $X$. Thanks to the Whitney embedding theorem and a standard approximation procedure, as a consequence, we give a new proof of the following well-known fact: if on an orientable, connected and compact $2$-dimensional smooth manifold there exists a continuous tangent vector field with no zeroes, then the manifold is diffeomorphic (or equivalently homeomorphic) to a torus.

math.DG↗
arXiv2019

On slow decay of Peetre's K-functional

We characterize when Peetre's K-functional decays to zero slowly and we use this characterization to demonstrate certain strict inclusions between real interpolation spaces.

math.FA↗
arXiv2017

On polynomial functions on non-conmmutative groups

Let $G$ be a topological group. We investigate relations between two classes of "polynomial like" continuous functions on $G$ defined, respectively, by the conditions (1) $Δ_h^{n+1}f=0$ for every $h \in G$, and (2) $Δ_{h_{n+1}} Δ_{h_{n}}\cdots Δ_{h_{1}}f=0$, for every $h_1,\cdots, h_{n+1} \in G$. It is shown that for many (but not all) groups these classes coincide. We consider also Montel type versions of the above conditions - when (1) and (2) hold only for steps $h$ in a generating subset of $G$. Our approach is based on the study of the counterparts of the discussed classes for general representations of groups (instead of the regular representation).

math.CA↗
arXiv2017

Characterization of distributions whose forward differences are exponential polynomials

Given $\{h_1,\cdots,h_{t}\} $ a finite subset of $\mathbb{R}^d$, we study the continuous complex valued functions and the Schwartz complex valued distributions $f$ defined on $\mathbb{R}^d$ with the property that the forward differences $Δ_{h_k}^{m_k}f$ are (in distributional sense) continuous exponential polynomials for some natural numbers $m_1,\cdots,m_t$.

math.CA↗
arXiv2017

Characterization of Polynomials as solutions of certain functional equations

Recently, the functional equation \[ \sum_{i=0}^mf_i(b_ix+c_iy)= \sum_{i=1}^na_i(y)v_i(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in\mathbf{GL}_d(\mathbb{C})$, was studied by Almira and Shulman, both in the classical context of continuous complex valued functions and in the framework of complex valued Schwartz distributions, where these equations were properly introduced in two different ways. The solution sets of these equations are, typically, exponential polynomials and, in some particular cases, they reduce to ordinary polynomials. In this paper we present several characterizations of ordinary polynomials as the solution sets of certain related functional equations. Some of these equations are important because of their connection with the Characterization Problem of distributions in Probability Theory.

math.CA↗
arXiv2017

On certain generalizations of the Levi-Civita and Wilson functional equations

We study the functional equation \[ \sum_{i=1}^mf_i(b_ix+c_iy)= \sum_{k=1}^nu_k(y)v_k(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in {GL}(d,\mathbb{R})$, both in the classical context of continuous complex-valued functions and in the framework of complex-valued Schwartz distributions, where these equations are properly introduced in two different ways. The solution sets are, typically, exponential polynomials and, in some particular cases, related to so called characterization problem of the normal distribution in Probability Theory, they reduce to ordinary polynomials.

math.CA↗
arXiv2016

An elementary inductive proof that $AB=I$ implies $BA=I$ for matrices

In this note we give an elementary demonstration of the fact that AB=I implies BA=I for square matrices A,B with coefficients in a field K. By elementary we mean that our proof follows from the very definitions of matrix and product of a matrix, with no extra help of more sophisticated results, as the use of dimensions of vector spaces or other ring- theoretical properties, like being Noetherian. The proof is also elementary in the sense that it relies on the concept and properties of the so called elementary operations on matrices. Finally, and no less important, the proof we show can be faced by any good student of a first year course in Mathematics, Physics or Engineering.

math.RA↗
arXiv2016

On Popoviciu-Ionescu functional equation

We study a functional equation first proposed by T. Popoviciu in 1955. It was solved for the easiest case by Ionescu in 1956 and, for the general case, by Ghiorcoiasiu and Roscau, and Radó in 1962. Our solution is based on a generalization of Radó's theorem to distributions in a higher dimensional setting and, as far as we know, is different than existing solutions. Finally, we propose several related open problems.

math.CA↗
arXiv2015

A dichotomy property for the graphs of monomials

We prove that the graph of a discontinuous $n$-monomial function $f:\mathbb{R}\to\mathbb{R}$ is either connected or totally disconnected. Furthermore, the discontinuous monomial functions with connected graph are characterized as those satisfying a certain big graph property. Finally, the connectedness properties of the graphs of additive functions $f:\mathbb{R}^d\to\mathbb{R}$ are studied.

math.CA↗