J. M. Almira

• Dr.
• Professor (Associate) at University of Murcia

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...

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 $$(A_0, A_1).$$ ( A 0 , A 1 ) . Lions problem, posed in 1960s, is to prove that different parameters $$(\theta ,p)$$ ( θ , p ) produc...

Talk presented at Aczel 100, International Scientific Conference celebrating the 100th birthday of J. Áczel, Hajdúszoboszlo, Hungary, February 02-07, 2025.

En esta charla, reflexiono ante un grupo de estudiantes de bachillerato, sobre la elección de la profesión de matemático, explicando las razones por las que yo mismo tuve esta vocación, de la que nunca me he arrepentido.

In this talk, I reflect before a group of high school students on the choice of the profession of mathematics, explaining the reasons why I myself had this vocation, which I have never regretted.

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 $(\theta,p)$ produce different interpolation spac...

Object detection is a main task in computer vision. Template matching is the reference method for detecting objects with arbitrary templates. However, template matching computational complexity depends on the rotation accuracy, being a limiting factor for large 3D images (tomograms). Here, we implement a new algorithm called tensorial template matc...

Normalized cross-correlation is the reference approach to carry out template matching on images. When it is computed in Fourier space, it can handle efficiently template translations but it cannot do so with template rotations. Including rotations requires sampling the whole space of rotations, repeating the computation of the correlation each time...

Normalized cross-correlation is the reference approach to carry out template matching on images. When it is computed in Fourier space, it can handle efficiently template translations but it cannot do so with template rotations. Including rotations requires sampling the whole space of rotations, repeating the computation of the correlation each time...

We present an inductive proof of the double-sidedness of the matrix inverse based on a property that holds true for associative rings with unity.

We consider Aichinger?s equation f (x1 +... + xm+1) = Xm+1 i=1 1i(x1, x2, ...,bxi,..., xm+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...

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,...

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,...

Se realiza un breve repaso de los teoremas de incompletitud de Gödel, con ánimo divulgativo.

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...

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...

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...

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.

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...

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...

A simple argument that uses Bayes Theorem is used to demonstrate the claim given by the tittle.

THIS IS VOL 6 OF "LA MATEMATICA CHE TRASFORMA IL MONDO"

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...

We propose an open problem connected to the characterization of certain spaces of measurable functions.

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...

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...

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...

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...

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.

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.

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.

Conferencia impartida en la Facultad de Matemáticas de la Universidad de Murcia

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.

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...

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...

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 $...

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 $...

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...

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...

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) $\Delta_h^{n+1}f=0$ for every $h \in G$, and (2) $\Delta_{h_{n+1}} \Delta_{h_{n}}\cdots \Delta_{h_{1}}f=0$, for every $h_1,\cdots, h_{n+1} \in G$. It is shown that for many (but n...

We prove a generalization of classical Montel's theorem for the mixed differences case, for polynomials and exponential polynomial functions, in commutative setting.

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...

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...

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...

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.

We give a new demonstration of Loewner's characterization of polynomials, solving in the positive a conjecture proposed by Laird and McCann in 1984.

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...

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...

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.

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.

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.

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...

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...

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...

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...

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...

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.

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...

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.

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...

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...

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...

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...

We study the finite dimensional spaces $V$ which are invariant under the action of the finite differences operator $\Delta_h^m$. Concretely, we prove that if $V$ is such an space, there exists a finite dimensional translation invariant space $W$ such that $V\subseteq W$. In particular, all elements of $V$ are exponential polynomials. Furthermore, $...

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...

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...

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 {@...

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.

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.

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.

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.

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.

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}.

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...

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.

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...

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...

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...

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...

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...

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...

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...

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...

In this paper, we concentrate our attention on the Muntz problem in the univariate setting and for the uniform norm.

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...

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.

In this paper, we concentrate our attention on the Müntz problem in the univariate setting and for the uniform norm.

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.

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...

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...