Conferencia originalmente ofrecida en septiembre de 1981 en Córdoba, Argentina, con motivo de la celebración del XXVº aniversario de la creación del Instituto de Matemática, Astronomía y Física de la Universidad Nacional de Córdoba.

Note: This text was not included in the print version of the Revista.

We characterize factor congruences in semilattices by using generalized notions of order ideal and of direct sum of ideals. When the semilattice has a minimum (maximum) element, these generalized ideals turn into ordinary (dual) ideals.

The bivariate confluent hypergeometric function kind 1 distribution is defined by the probability density function proportional to x₁^{ν₁ − 1} x₂^{ν₂ − 1}₁F₁(α; β; −x₁ − x₂). In this article, we study several properties of this distribution and derive density functions of X₁/X₂, X₁/(X₁ + X₂), X₁ + X₂ and 2 √(X₁ X₂). The density function of 2 √(X₁ X₂) is represented in terms of modified Bessel function of the second kind. We also show that for ν₁ − ν₂ = 1/2, 2 √(X₁ X₂) follows a confluent hypergeometric function kind 1 distribution.

Let T be a bounded linear operator on a Banach space X. We prove some properties of X₁ = { z ∈ X : lim_n ∑_k=1ⁿ (T^kz/k) exists} and we construct an operator T such that lim_n ||Tⁿ/n|| = 0, but (I − T)X is not included in X₁.

In the present paper, we have proved a general summability factor theorem by using a general summability method. This theorem also includes several new results.

We find the explicit parametric equations of the flat 3-dimensional integral C-parallel submanifolds in the sphere S⁷ endowed with the deformed Sasakian structure defined by Tanno.

We define a local type of a weighted BMO space on R⁺ and prove the boundedness of the local Hardy-Littlewood maximal function in that space, provided that the weight belongs to the local class A¹_loc.

We present a new approach to prove global L^p estimates for degenerate Ornstein-Uhlenbeck operators in R^N. We then show how to pave the way to extend such a technique to classes of general Hörmander operators. Several historical notes related to Calderón-Zygmund’s singular integrals theory in Euclidean and in non-Euclidean settings are also provided.

The present paper is devoted to study the algebraic sets of normal sections of the so called Cartan’s isoparametric hypersurfaces M_R, M_C, M_H and M_O of complete flags in the projective planes RP², CP², HP² and OP². It presents a connection between “normed trialities” and the polynomial defining the algebraic sets of normal sections of these hypersurfaces. It contains also a geometric and topological description of these algebraic sets by means of the isotropy actions of the isoparametric hypersurfaces.

Starting from the study of the Shepard nonlinear operator of max-prod type in [2], [3], in the recent monograph [4], Open Problem 5.5.4, pp. 324-326, the Baskakov max-prod type operator is introduced and the question of the approximation order by this operator is raised. The aim of this note is to obtain the order of uniform approximation Cω₁(f; 1/√n) (with the explicit constant C = 24) of another operator called the truncated max-prod Baskakov operator and to prove by a counterexample that in some sense, for arbitrary f this type of order of approximation with respect to ω₁(f; √n) cannot be improved. However, for some subclasses of functions including for example the nondecreasing concave functions, the essentially better order of approximation ω₁(f; 1/n) is obtained. Finally, some shape preserving properties are proved.

Some inequalities of the Slater type for convex functions of selfadjoint operators in Hilbert spaces $H$ under suitable assumptions for the involved operators are given. Amongst others, it is shown that if $A$ is a positive definite operator with $Sp(A) \subset [m,M]$ and $f$ is convex and has a continuous derivative on $[m,M]$, then for any $x\in H$ with $\left\Vert x\right\Vert =1$ the following inequality holds: \begin{multline*} 0\leq f\left( \frac{\left\langle Af^{\prime }\left( A\right) x,x\right\rangle }{\left\langle f^{\prime }\left( A\right) x,x\right\rangle }% \right) -\left\langle f\left( A\right) x,x\right\rangle \\ \leq \frac{1}{4}\cdot \sqrt{\frac{Mf^{\prime }\left( M\right) }{mf^{\prime }\left( m\right) }}\left( M-m\right) \left( f^{\prime }\left( M\right) -f^{\prime }\left( m\right) \right). \end{multline*}

In this paper, we prove that the extended values $E(r,s;x,y)$ are Schur harmonic convex (or concave, respectively) with respect to $(x,y)\in (0,\infty)\times(0,\infty)$ if and only if $(r,s)\in \{(r,s):s\geq-1, s\geq r,s+r+3\geq0\}\cup\{(r,s):r\geq-1, r\geq s,s+r+3\geq0\}$ (or $\{(r,s):s\leq-1, r\leq-1,s+r+3\leq0\},$ respectively).

In this paper, we prove the following assertions: (i) If T is of quasi-class (A, k), then T is polaroid and reguloid; (ii) If T or T^* is an algebraically of quasi-class (A, k) operator, then Weyls theorem holds for f(T) for every f ∈ Hol(σ(T)); (iii) If T^* is an algebraically of quasi-class (A, k) operator, then a-Weyls theorem holds for f(T) for every f ∈ Hol(σ(T)); (iv) If T^* is algebraically of quasi-class (A, k) then property (ω) holds for T.

In this paper, we apply He’s polynomials for finding the approximate solution of the Troesch’s problem which arises in the confinement of a plasma column by radiation pressure and applied physics. The proposed technique proved to be very effective and is easier to implement as compare to decomposition method.

This paper is a non exhaustive survey about Hardy spaces defined by semigroups of operators.

This work attempts to be an overview of a variety of results concerning Sobolev spaces associated to some orthonormal systems, particularly the Hermite and Laguerre operators settings.

We give the symmetric version of five lemmas which are essential for the theory of tensor products (and norms). These are: the approximation, extension, embedding, density and local technique lemma. Some applications of these tools to the metric theory of symmetric tensor products and to the theory of polynomials ideals are given.

In this expository paper we present various general extensions of the celebrated Wiener’s Tauberian Lemma, outline several ingredients that are often used in proofs and discuss a few applications to localization of frames.

Porosity and dimension are two useful, but different, concepts that quantify the size of fractal sets and measures. An active area of research concerns understanding the relationship between these two concepts. In this article we will survey the various notions of porosity of sets and measures that have been proposed, and how they relate to dimension. Along the way, we will introduce the idea of local entropy averages, which arose in a different context, and was then applied to obtain a bound for the dimension of mean porous measures.

In engineering and applied mathematics, Zak transforms have been effectively used for over 50 years in various applied settings. As Gelfand observed in a 1950 paper, the variable coefficient Fourier series ideas articulated in Andre Weil’s famous book on integration lead to an exceedingly elementary proof of the Plancherel Theorem for LCA groups. The transform for functions on R appearing in Zak’s seminal 1967 paper is actually a special case of the LCA group transforms earlier introduced by Weil; Zak states this explicitly in his 1967 paper but the mathematical community nonetheless chose to name the transform for him.

In brief, the properties of Zak transforms are simply reflections of elementary Fourier series properties and the Plancherel Theorem for non-compact LCA groups is an immediate consequence of the fact that Fourier transforms are averages of Zak transforms. It is remarkable that only a small handful of mathematicians know this proof and that all textbooks continue to give much harder and less transparent proofs for even the case of the group R. Generalized Zak transforms arise naturally as intertwining operators for various representations of Abelian groups and allow formulation of many appealing theorems.