J. M. Almira

J. M. Almira
University of Murcia | UM · Depart. Ing. y Tecn. Comp. ´

Dr.

About

115
Publications
13,225
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
396
Citations
Introduction
J. M. Almira currently works at the Depart. Ing. y Tecn. Comp. ´, University of Murcia. J. does research in Analysis and Applied Mathematics although his main interest is actually focused on Functional Equations. His most recent publication is 'On polynomial functions on non-commutative groups'.
Additional affiliations
September 1999 - present
Universidad de Jaén
Position
  • Professor (Associate)

Publications

Publications (115)
Preprint
Full-text available
We prove, for d>1, a negative result for approximation of functions defined con compact subsets of R^d with single layer feedforward neural networks with arbitrary activation functions. In philosophical terms, this result claims the existence of learning functions f(x) which are as difficult to approximate with these neural networks as one may want...
Conference Paper
Full-text available
Se realiza un breve repaso de los teoremas de incompletitud de Gödel, con ánimo divulgativo.
Preprint
Full-text available
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 poly...
Preprint
Full-text available
If f(z) is either a polynomial with no zeroes or a bounded entire function, then a Riemannian metric g_f is constructed on the complex plane C. This metric g_f is shown to be flat and geodesically complete. Therefore, the Riemannian manifold (C, g_f) must be isometric to (C, |dz|^2), which implies that f(z) is a constant. MSC2020: 30C15, 30D20, 53...
Book
Thanks to the so-called Uniform Sampling Theorem, all the information we need to preserve or transmit between physical devices can be digital. This is the main reason why digital communication has invaded our lives in recent decades. To understand this digitization of the world, in addition to the sampling theorem, it is essential to have a mathema...
Article
We characterize when Peetre's K-functional slowly decays to zero and we use this characterization to demonstrate certain strict inclusions between real interpolation spaces.
Article
We prove a negative result for the approximation of functions defined on compact subsets of Rd (where d≥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 m...
Book
Nel secondo decennio del XXI secolo ci siamo abituati a parlare con assistenti virtuali come Siri, Google Assistant, Cortana o Alexa. Per il momento formuliamo domande e diamo ordini usando frasi semplici e brevi, aspettandoci da loro un'unica risposta. E' prevedibile che in futuro, però, queste interazioni siano destinate a diventare sempre più co...
Preprint
Full-text available
A simple argument that uses Bayes Theorem is used to demonstrate the claim given by the tittle.
Book
THIS IS VOL 6 OF "LA MATEMATICA CHE TRASFORMA IL MONDO"
Book
How many degrees of separation are there between Facebook users? How to know who is an influencer on Instagram? How does viral news spread on Twitter? A mathematical discipline born almost three centuries ago seeks to answer these questions, as well as study and understand the formation of modern social networks and their evolution: the theory of g...
Presentation
Full-text available
We propose an open problem connected to the characterization of certain spaces of measurable functions.
Article
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 gen...
Preprint
Full-text available
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 Rad\'{o} in 1962. We present a solution to the equation both for the one-dimensional and the higher-dimensional cases, which is based on a...
Preprint
Full-text available
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 app...
Preprint
Full-text available
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 app...
Preprint
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.
Article
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.
Presentation
Full-text available
Transparencias de una conferencia impartida por invitación de la Academia de Ciencias de Granada el 27/03/2019, en la Facultad de Ciencias de la Universidad de Granada.
Presentation
Full-text available
Conferencia impartida en la Facultad de Matemáticas de la Universidad de Murcia
Article
We show that in the existing bibliography there are many contradictory claims about the exact date of death of Hermann Weyl and, after a detailed exposition of the way we got our evidence, we demonstrate that Weyl died on 8 December 1955. The paper also shows up the fact that an authoritative intellectual biography on Weyl is yet to be written.
Article
Full-text available
We study the functional equation (Formula presented.)with (Formula presented.) and (Formula presented.), 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, exponentia...
Article
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) δhn+1f=0 for every h∈G, and 2) δhn+1δhn⋯δh1f=0 for every h1,⋯,hn+1∈G. It is shown that for many (but not all) groups these classes coincide. We consider also Montel type versions of th...
Article
Full-text available
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 $\Delta_{h_k}^{m_k}f$ are (in distributional sense) continuous exponential polynomials for some natural numbers $...
Article
Full-text available
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...
Article
Full-text available
We initiate a study of three different classes of functions defined on a general group: polynomials, semipolynomials and quasipolynomials. Our main aim is twofold. On the one hand, we study the existing inclusion relations between these classes and, on the other hand, we study Montel type theorems in this context. Also, we present a new proof of cl...
Article
Full-text available
We prove a generalization of classical Montel's theorem for the mixed differences case, for polynomials and exponential polynomial functions, in commutative setting.
Article
Full-text available
We study 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 $b_i,c_i\in\mathbf{GL}_d(\mathbb{C})$, 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 diff...
Book
Full-text available
El cerebro es, sin duda, el sistema más complejo al que podemos enfrentarnos. Su actividad eléctrica, que se produce con amplitud de micro- voltios, se recoge mediante el uso de electrodos repartidos homogéneamente desde la superficie exterior del cráneo, dando lugar a los llamados electroencefalogramas (EEG). Se trata de potenciales extremadamente...
Article
Full-text available
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 rin...
Article
Full-text available
We give a new demonstration of Loewner's characterization of polynomials, solving in the positive a conjecture proposed by Laird and McCann in 1984.
Article
Full-text available
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 Rad\'o in 1962. Our solution is based on a generalization of Rad\'o's theorem to distributions in a higher dimensional setting and, as far as we know, is different th...
Article
Full-text available
Assume that a linear space of real polynomials in $n$ variables is given which is translation and dilation invariant. We show that if a sequence in this space converges pointwise to a polynomial, then the limit polynomial belongs to the space, too.
Article
Full-text available
Assume that a linear space of real polynomials in n variables is given which is translation and dilation invariant. We show that if a sequence in this space converges pointwise to a polynomial, then the limit polynomial belongs to the space, too.
Article
Full-text available
This paper is an introduction to the regularity theory of functional equations, motivated by the study of Fréchet's functional equation. Another main goal is to honor the work in functional equations of the Romanian mathematician Tiberiu Popoviciu.
Article
Full-text available
Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R, C) to give a new proof of classical Montel’s theorem, about continuous solutions of Frechet’s functional equation ∆m h f = 0, for real functions (and complex functions) of one real variable. In this paper we use similar ideas t...
Article
Full-text available
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...
Article
Full-text available
In this paper we characterize local exponential monomials and polynomials on different types of Abelian groups and we prove Montel-type theorems for these function classes.
Article
Full-text available
We present an elementary proof of a general version of Montel's theorem in several variables which is based on the use of tensor product polynomial interpolation. We also prove a Montel-Popoviciu's type theorem for functions $f:\mathbb{R}^d\to\mathbb{R}$ for $d>1$. Furthermore, our proof of this result is also valid for the case $d=1$, differing in...
Data
We prove a version of Montel's Theorem for the case of continuous functions defined over the field Q(p) of p-adic numbers. In particular, we prove that, if Delta(m+1)(h0)f(x) = 0 for all x is an element of Q(p), and h(0) satisfies vertical bar h(0)vertical bar(p) = p (N0), then, for all x(0) is an element of Q(p), the restriction of f over the set...
Article
Full-text available
In this paper some classes of local polynomial functions on abelian groups are characterized by the properties of their variety. For this characterization we introduce a numerical quantity depending on the variety of the local polynomial only. Moreover, we show that the known characterization of polynomials among generalized polynomials can be simp...
Article
Full-text available
In this paper local polynomials on Abelian groups are characterized by a "local" Fr\'echet-type functional equation. We apply our result to generalize Montel's Theorem and to obtain Montel-type theorems on commutative groups.
Book
J. B. J. Fourier (1768-1830), autor de la Teoría analítica del calor, es considerado uno de los fundadores de la física matemática moderna. Al tratar un tema que, por su propia naturaleza, quedaba fuera de los dominios de la mecánica racional, se vio obligado a enfrentarse a ciertos problemas completamente nuevos, y los abordó con más imaginación...
Article
Full-text available
We prove that, if f:R^n\to R satisfies Fr\'echet's functional equation and f(x_1,...,x_n) is not an ordinary algebraic polynomial in the variables x_1,...,x_n, then f is unbounded on all non-empty open set U of R^n. Furthermore, the closure of its graph contains an unbounded open set.
Article
Full-text available
We present an overview of some results about characterization of compactness in which the concept of approximation scheme has had a role. In particular, we present several results that were proved by the second author, jointly with Luther, a decade ago, when these authors were working on a very general theory of approximation spaces. We then introd...
Article
Full-text available
We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of continuous solutions of certain equations are closed vector subspaces of $C(\mathbb{C}^d,\mathbb{C})$ which a...
Article
Full-text available
Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R,C) to give a new proof of classical Montel's theorem, about continuous solutions of Fr\'{e}chet's functional equation $\Delta_h^mf=0$, for real functions (and complex functions) of one real variable. In this paper we use similar...
Article
Full-text available
We study the finite-dimensional spaces V, that are invariant under the action of the finite differences operator . Concretely, we prove that if V is such an space, there exists a finite-dimensional translation invariant space W such that V ⊆ W. In particular, all elements of V are exponential polynomials. Furthermore, V admits a decomposition V = P...
Article
Full-text available
We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if \[ \Delta_{h_0}^{m+1}f(x)=0 \ \ \text{for all} x\in\mathbb{Q}_p, \] and $|h_0|_p=p^{-N_0}$ then, for all $x_0\in \mathbb{Q}_p$, the restriction of $f$ over the set...
Article
Full-text available
Shapiro's lethargy theorem states that if {A_n} is any non-trivial linear approximation scheme on a Banach space X, then the sequences of errors of best approximation E(x,A_n) = \inf_{a \in A_n} ||x - a_n||_X decay almost arbitrarily slowly. Recently, Almira and Oikhberg investigated this kind of result for general approximation schemes in the quas...
Article
An approximation scheme is a family of homogeneous subsets (A"n) of a quasi-Banach space X, such that A"1@?A"2@?...@?X, A"n+A"n@?A"K"("n"), and @?"nA"n@?=X. Continuing the line of research originating at the classical paper [8] by Bernstein, we give several characterizations of the approximation schemes with the property that, for every sequence {@...
Article
Full-text available
We study discontinuous solutions of the monomial equation $\frac{1}{n!}\Delta_h^nf(x)=f(h)$. In particular, we characterize the closure of their graph, $\bar{G(f)}^{\mathbb{R}^2}$, and we use the properties of these functions to present a new proof of the Darboux type theorem for polynomials and of Hamel's theorem for additive functions.
Article
If a (non-constant) polynomial has no zero, then a certain Riemannian metric is constructed on the two-dimensional sphere. Several geometric arguments are then shown to contradict this fact.
Article
Full-text available
We study some properties of the solutions of the functional equation $$f(x)+f(a_1x)+...+f(a_Nx)=0,$$ which was introduced in the literature by Mora, Cherruault and Ziadi in 1999, for the case $a_k=k+1$, $k=1,2,...,N$ and studied by Mora in 2008 and Mora and Sepulcre in 2009 and 2011.
Article
Full-text available
In this paper we show a lethargy result in the non-Arquimedian context, for general ultrametric approximation schemes and, as a consequence, we prove the existence of p-adic transcendental numbers whose best approximation errors by algebraic p-adic numbers of degree less than or equal to n decays slowly.
Article
Full-text available
If a (non-constant) polynomial has no zero, then a certain Riemannian metric is constructed on the two dimensional sphere. Several geometric arguments are then shown to contradict this fact.
Article
Full-text available
Given X,Y two Q-vector spaces, and f:X -> Y, we study under which conditions on the sets $B_k\subseteq X$, k=1,...,s, if $\Delta_{h_1h_2... h_s}f(x)=0$ for all x in X and h_k in B_k, k=1,2,...,s, then $\Delta_{h_1h_2... h_s}f(x)=0$ for all (x,h_1,...,h_s) in X^{s+1}.
Article
Full-text available
Approximation spaces, in their many presentations, are well known mathematical objects and many authors have studied them for long time. They were introduced by P. L. Butzer and K. Scherer [Approximationsprozesse und Interpolationsmethoden. Mannheim-Zürich: Bibliographisches Institut (1968; Zbl 0177.08501)] in 1968 and, independently, by Ju. A. Bru...
Article
Full-text available
In this paper we introduce two digital zoom methods based on sampling theory and we study their mathematical foundation. The first one (usu-ally known by the names of 'sinc interpolation', 'zero-padding' and 'Fourier zoom') is commonly used by the image processing community.
Article
In a previous paper (see arXiv:1003.3411 [math.CA]), we investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (A_n) (defined by E(x,A_n) = \inf_{a \in A_n} \|x - a_n\|) decay arbitrarily slowly. In this work, we consider the question of whether x witnessing the slowne...
Article
Full-text available
Let $X$ be a Banach space and suppose $Y\subseteq X$ is a Banach space compactly embedded into $X$, and $(a_k)$ is a weakly null sequence of functionals in $X^*$. Then there exists a sequence $\{\varepsilon_n\} \searrow 0$ such that $|a_n(y)| \leq \varepsilon_n \|y\|_Y$ for every $n\in\mathbb{N}$ and every $y\in Y$. We prove this result and we use...
Article
In this short note we prove that, if (C[a,b],{A_n}) is an approximation scheme and (A_n) satisfies de La Vall\'ee-Poussin Theorem, there are instances of continuous functions on [a,b], real analytic on (a,b], which are poorly approximable by the elements of the approximation scheme (A_n). This illustrates the thesis that the smoothness conditions g...
Article
The main goal of this note is to prove that, in general, the smoothness concepts derived from membership to an approximation space, are of global nature. To prove this claim we show that if (C[a,b],{An}) is an approximation scheme and (An) satisfies de La Vall\'ee-Poussin Theorem, there are very smooth functions (in the classical sense) failing the...
Article
Full-text available
In this paper we characterize the approximation schemes that satisfy Shapiro's theorem and we use this result for several classical approximation processes. In particular, we study approximation of operators by finite rank operators and n-term approximation for several dictionaries and norms. Moreover, we compare our main theorem with a classical r...
Conference Paper
Full-text available
In this note we explain the main motivations Norbert Wiener had for the creation of his Generalized Harmonic Analysis [13] and his Tauberian Theorems [14]. Although these papers belong to the most pure mathematical tradition, they were deeply based on some Engineering and Physics Problems and Wiener was able to use them for such diverse areas as Op...
Book
Norbert Wiener (1894-1964) fue uno de los primeros matemáticos estadounidenses que alcanzó prestigio internacional; su investigación, que siempre estuvo motivada por la física, la ingeniería o la biología, tiene el sabor de una curiosa mezcla de análisis de Fourier y teoría de la probabilidad. Sus contribuciones al estudio de las corrientes débiles...
Article
Full-text available
In this paper the characterization as convolution operators of filters sending finite energy signals to bounded signals is used to prove several theoretical results concerning the distance between the ideal filter and the spaces of physically realizable filters. Both the analog and the digital cases are studied and the formulas for the distance and...
Article
Full-text available
In this paper, we concentrate our attention on the Muntz problem in the univariate setting and for the uniform norm.
Article
In this paper we give a new proof of a classical result by Fréchet [M. Fréchet, Une définition fonctionnelle des polynomes, Nouv. Ann. 9 (4) (1909) 145–162]. Concretely, we prove that, if and f is continuous at some point or bounded at some nonempty open set, then f∈Pk. Moreover, as a consequence of the technique developed for our proof, it is poss...
Article
Full-text available
The first geometric proof of the Fundamental Theorem of Algebra is given. It is proved from the (abstract) Gauss-Bonnet theorem for the 2-sphere.
Article
Full-text available
In this paper, we concentrate our attention on the Müntz problem in the univariate setting and for the uniform norm.
Article
We give a new proof of Hilbert’s Nullstellensatz, based on the use of Gröbner basis. The proof has two variants. The first one uses the fundamental theorem of algebra and the second one uses Gelfand-Mazur’s theorem.
Book
La obra de David Hilbert constituye un legado de incalculable valor sin el cual no se explica la matemática del siglo XX. Su fecunda carrera comenzó en 1888 con la espectacular resolución del problema de Gordan, una de las cuestiones más esquivas de la época. En su perfil científico despuntaron de inmediato la enorme capacidad de trabajo, una penet...
Article
Full-text available
We show that generalized approximation spaces can be used to describe the relatively compact sets of Banach spaces. This leads to compactness and convergence criteria in the approximation spaces themselves. If these spaces can be described with the help of moduli of smoothness, then the criteria can be formulated in terms of the moduli. As applicat...
Article
Full-text available
We solve the Müntz problem for the space ℂ(K) whenever K⊂[0,∞) is a countable compact set which satisfies certain additional assumptions and we propose the general case as an open question.
Article
We prove the inverse closedness of certain approximation algebras based on a quasi-Banach algebra X using two general theorems on the inverse closedness of subspaces of quasi-Banach algebras. In the first theorem commutative algebras are considered while the second theorem can be applied to arbitrary X and to subspaces of X which can be obtained by...
Article
Full-text available
We prove the existence of a dense subset Δ of [ 0 , 4 ] such that for all α ∈ Δ there exists a subgroup X α of infinite rank of ℤ [ z ] such that X α is a discrete subgroup of C [ 0 , β ] for all β ≥ α but it is not a discrete subgroup of C [ 0 , β ] for any β ∈ ( 0 , α ) . Given a set of nonnegative real numbers Λ = { λ i } i = 0 ∞ , a Λ -polynomi...
Article
Full-text available
In this paper we study the stability character of the linear differential equation x′ = Ax, where A = A(t) is a piecewise constant matrix function of period T > 0, with A(t) ∈ Cn for all t and a fixed class Cn of matrices of order n. Concretely, we are interested in the characterization of the permutations of the pieces of A which do not change the...
Article
The main goal of this note is to give a new elementary proof of the Fundamental Theorem of Algebra,. This proof is based on the use of the well known Gelfand-Mazur Theorem.
Article
In this note we prove several Müntz-type results for Simultaneous Diophantine Approximation on intervals of length smaller than 2√2.
Article
In this note we give an easy proof of a Müntz-type theorem for the approximation of functions by polynomials with integral coefficients on arbitrary intervals of length smaller than four.
Article
Full-text available
The main goal of this paper is to put some light in several arguments that have been used through the time in many contexts of Best Approximation Theory to produce proximinality results. In all these works, the main idea was to prove that the sets we are considering have certain properties which are very near to the compactness in the usual sense....
Article
Full-text available
In this paper we use a generalized theory of approximation spaces to prove several of the so called negative results in Approximation Theory. Concretely, we give new proofs of the classical lethargy Theorems.
Article
In the paper we generalize the theory of classical approximation spaces to a much wider class of spaces which are defined with the help of best approximation errors. We also give some applications. For example, we show that generalized approximation spaces can be used to find natural (in some sense) domains of definition of unbounded operators. (©...
Article
Full-text available
In this paper, we continue some work by Canada and Drabek [1] and Mawhin [6] on the range of the Neumann and Periodic boundary value problems:egin{gather*} mathbf{u}''(t)+mathbf{g}(t,mathbf{u}'(t))= overline{mathbf{f}}+widetilde{mathbf{f}}(t), quad tin (a,b) mathbf{u}'(a)=mathbf{u}'(b)=0 ext{or}quad mathbf{u}(a)=mathbf{u}(b),quad mathbf{u}'(a)=math...
Article
In this note we study the following question: Under which minimal assumptions on the function f is it true that the condition (ΔN+1h f)(a) = 0 for all a ∈ R and h ≥ 0 implies that f is a polynomial of degree less than or equal to N?
Article
Full-text available
In this note, we give several definitions of metric corona properties which could be of interest in Set Topology, Functional Analysis and Approximation Theory, and prove that there are complete metrizable t.v.s. which are nice in the sense that they have a metric which is invariant by translations, but they do not have good corona properties. All c...