Antonio Galbis

Nonradial Hörmander algebras of several variables and convolution operators

A characterization of the closed principal ideals in nonradial Hormander algebras of holomorphic functions of several variables in terms of the behaviour of the generator is obtained. This result is applied to study the range of convolution operators and ultradifferential operators on spaces of quasianalytic functions of Beurling type. Contrary to what is known to happen in the case of non-quasianalytic functions, an ultradistribution on a space of quasianalytic functions is constructed such that the range of the operator does not contain the real analytic functions. Let u, v : R → R be continuous, non-negative and even functions which are increasing on the positive real numbers. We assume that v is convex and the quotient u(x) v(x) tends to zero as x → ∞. Both functions are extended to R as follows: u(x1, . . . , xN ) := N ∑ i=1 u(xi), v(x1, . . . , xN ) := N ∑

On Taylor Coefficients of Entire Functions Integrable against Exponential Weights

In this paper we shall analyze the Taylor coefficients of entire functions integrable against dμp(z) = p 2π e−|z| p | z |p−2 dσ(z) where dσ stands for the Lebesgue measure on the plane and p ∈ IN, as well as the Taylor coefficients of entire functions in some weighted sup-norm spaces. In this paper we shall analyze the Taylor coefficients of entire functions satisfying some growth estimates. To be more precise, given p ∈ IN, we will deal with the Banach space B1(p) of entire functions belonging to L1(dμ), where dμ(z) = p 2π e −|z| | z |p−2 dσ(z) and dσ stands for the Lebesgue measure on the plane, as well as with the Banach space H(e−|z| p )(C) of those entire functions f such that supz∈C e−|z| p | f(z) |< ∞. These spaces have been considered in several contexts by different authors. See [1, 6, 7, 8, 9, 10]. The general question we are going to discuss can be stated as follows: given a function f(z) = ∑∞ n=0 anz n in X(:= B1(p) or H(e−|z| p )(C)), what can be said on the Taylor coefficients (an)?. Conversely, it is also interesting to ask how a function in X can be recognized by the behaviour of its Taylor coefficients. The paper is organized as follows. In the first section we present a method to describe the boundedness of operators from B1(p) into a general Banach space X by the fact that the X−valued analytic function constructed by the action of the operator on the reproducing kernel Kp belongs to the vector-valued space H(e−|z| p )(C;X). This will allow to identify the dual space of B1(p) with the weighted sup-norm space H(e−|z| p )(C). Then we will discuss a Hardy’s type inequality for Taylor coefficients of functions in B1(p). In the second section we give a complete characterization of the Taylor coefficients for lacunary entire functions in both spaces B1(p) and H(e−|z| p )(C). As an application we obtain a sufficient condition on the Taylor coefficients of a function f in order to enssure that it belongs to H(e−|z| p )(C). In section 3 we find conditions on nk in order to get the unconditional convergence of ∑ akz nk to be equivalent to the absolute convergence of the series. Let us denote by H(e−|z| p )0(C) the closed subspace of H(e−|z| p )(C) consisting of those functions f such that e−|z| p f(z) vanishes at infinity. Since

Tensor products of Fréchet or (DF)-spaces with a Banach space

Abstract The aim of the present article is to study the projective tensor product of a Frechet space and a Banach space and the injective tensor product of a (DF)-space and a Banach space. The main purpose is to analyze the connection of the good behaviour of the bounded subsets of the projective tensor product and of the locally convex structure of the injective tensor product with the local structure of the Banach space.

Beurling ultradistributions of Lp-growth

We study the convolutors and the surjective convolution operators acting on spaces of ultradistributions of Lp-growth. In the case p = 2 we obtain complete characterizations. Some results on hypoellipticity are also included.  2003 Elsevier Science (USA). All rights reserved.

Atrial fibrillation subtypes classification using the General Fourier-family Transform

Atrial fibrillation patients can be classified into paroxysmal, persistent and permanent attending to the temporal pattern of this arrhythmia. The surface electrocardiogram hides this differentiation. A classification method to discriminate between the different subtypes of atrial fibrillation by using short segments of electrocardiograms recordings is presented. We will process the electrocardiograms (ECGs) using time-frequency techniques with a global accuracy of 80%. Real cases are evaluated showing promising results for an implementation in a semiautomated diagnostic system.

A NOTE ON TASKINEN'S COUNTEREXAMPLES ON THE PROBLEM OF TOPOLOGIES OF GROTHENDIECK

By the work of Taskinen (see [4, 5]), we know that there is a Frechet space E such that L b ( E , l 2 ) is not a ( DF )-space. Moreover there is a Frechet–Montel space F such that is not ( DF ). In this second example, the duality theorem of Buchwalter (cf. [2, §45.3]) can be applied to obtain that and hence is a ( gDF )-space (cf. [1, Ch. 12 or 3, Ch. 8]). The ( gDF )-spaces were introduced by several authors to extend the ( DF )-spaces of Grothendieck and to provide an adequate frame to consider strict topologies.

Surfaces with Boundary

One of the objectives of this book is to obtain a rigorous proof of a version of Green’s formula for compact subsets of \(\mathbb{R}^2\) whose topological boundary is a regular curve of class C 2. These sets are typical examples of what we will call regular 2-surfaces with boundary in \(\mathbb{R}^2\). The analogous three-dimensional example would consist of a compact set of \(\mathbb{R}^3\) whose topological boundary is a regular surface of class C 2. The following example is perhaps instructive.

Some remarks on compact Weyl operators

We show that, if σ∈𝒮′(ℝ2d ) is a tempered distribution and, for some 1<p, q<∞, the Weyl operator L σ acts as a compact operator L σ:M p,q (ℝ d )→M p,q (ℝ d ), then the short time Fourier transform of σ vanishes at infinity. Some results on the eigenvalue distributions of the operators are included, as well as an example of a non-zero function with constant modulus, which is the symbol of a compact localization operator.

Perturbations of surjective convolution operators

Let μ 1 and μ 2 be (ultra)distributions with compact support which have disjoint singular supports. We assume that the convolution operator f → μ 1 *f is surjective when it acts on a space of functions or (ultra)distributions, and we investigate whether the perturbed convolution operator f→ (μ 1 + μ 2 ) * f is surjective. In particular we solve in the negative a question asked by Abramczuk in 1984.

Localization Operators and an Uncertainty Principle for the Discrete Short Time Fourier Transform

Localization operators in the discrete setting are used to obtain information on a signal f from the knowledge on the support of its short time Fourier transform. In particular, the extremal functions of the uncertainty principle for the discrete short time Fourier transform are characterized and their connection with functions that generate a time-frequency basis is studied.