J. M. Almira

Compactness and Generalized Approximation Spaces

ABSTRACT 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 applications we give a generalization of Bernsteins theorem about existence of elements with prescribed best approximation errors, compactness criteria for operators, a criterion for compactness in Sobolev type spaces, and a generalization of Simons compactness criterion for subsets of Lp -spaces of Banach-space-valued functions.

Montel's Theorem and Subspaces of Distributions Which Are Δ m -Invariant

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 ⊕ E with P a space of polynomials and E a translation invariant space. As a consequence of this study, we prove a generalization of a famous result by Montel [7], which states that, if f: ℝ → ℂ is a continuous function satisfying for all t ∈ ℝ and certain h 1, h 2 ∈ ℝ∖{0} such that h 1/h 2 ∉ ℚ, then f(t) = a 0 + a 1 t + … +a m−1 t m−1 for all t ∈ ℝ and certain complex numbers a 0, a 1,…, a m−1. We demonstrate, with quite different arguments, the same result not only for ordinary functions f(t) but also for complex valued distributions. Finally, we also consider the subspaces V that are Δ h 1 h 2…h m -invariant for all h 1,…, h m ∈ ℝ.

Local polynomials and the Montel theorem

In this paper local polynomials on Abelian groups are characterized by a “local” Fréchet-type functional equation. We apply our result to generalize Montel’s Theorem and to obtain Montel-type theorems on commutative groups.

On Popoviciu-Ionescu Functional Equation

Abstract We study a functional equation first proposed by T. Popoviciu [15] in 1955. It was solved for the easiest case by Ionescu [9] in 1956 and, for the general case, by Ghiorcoiasiu and Roscau [7] and Radó [17] 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.

A lethargy result for real analytic functions

Abstract We prove that, if ( C [ a , b ] , { A n } ) is an approximation scheme and { A n } satisfies the de La Vallee Poussin Theorem, there are examples of real-valued continuous functions on [ a , b ] , analytic on ( a , b ] , which are “poorly approximable” by the elements of { A n } . This illustrates the thesis that the smoothness conditions guaranteeing that a function is “well approximable” must be “global”. The failure of smoothness at endpoints may result in an arbitrarily slow rate of approximation.

Another topological proof of the Fundamental Theorem of Algebra

. Allen Lesern wird der Fundamentalsatz der Algebra bekannt sein. Er besagt, dass jedes Polynom P = P(z) uber dem Korper der komplexen Zahlen mindestens eine komplexe Nullstelle hat. Besitzt P den Grad n, so ergibt sich daraus sofort, dass P (mit Vielfachheiten gezahlt) genau n komplexe Nullstellen hat. Die Bestimmung der Nullstellen von Polynomen spielte in der Entwicklung der Algebra eine wichtige Rolle. Allerdings gelang es erst N.H. Abel zu beweisen, dass die Nullstellen eines Polynoms vom Grad n > 4 in der Regel nicht durch Radikale darstellbar sind. Damit musste zum Beweis des Fundamentalsatzes nach neuen Ideen gesucht werden. Neben den Beweisen von C.F. Gauss wird der Fundamentalsatz heute sehr oft als elegante Anwendung aus dem Satz von Liouville in der Funktionentheorie gefolgert. Im vorliegenden Beitrag geben die Autoren einen ebenfalls eleganten Beweis des Fundamentalsatzes, der auf einfachen Ergebnissen der Topologie beruht.

A note on Diophantine approximation

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 -polynomial (or Muntz polynomial) is a function of the form p ( x ) = ∑ i = 0 n a i z λ i ( n ∈ ℕ ). We denote by Π ( Λ ) the space of Λ -polynomials and by Π ℤ ( Λ ) : = { p ( x ) = ∑ i = 0 n a i z λ i ∈ Π ( λ ) : a i ∈ ℤ  for all  i ≥ 0 } the set of integral Λ -polynomials. Clearly, the sets Π ℤ ( Λ ) are subgroups of infinite rank of ℤ [ x ] whenever Λ ⊂ ℕ , # Λ = ∞ (by infinite rank, we mean that the real vector space spanned by X does not have finite dimension. In all what follows we are uniquely interested in groups of infinite rank). Now, it is well known that the problem of approximation of functions on intervals [ a , b ] by polynomials with integral coefficients is solvable only for intervals [ a , b ] of length smaller than four and functions f which are interpolable by polynomials of ℤ [ x ] on a certain set (which we call the algebraic kernel of the interval [ a , b ] ) 𝒥  ( a , b ) . Concretely, it is well known that ℤ [ x ] is a discrete subgroup of C [ a , b ] whenever b − a ≥ 4 and 4 is the smallest number with this property (for these and other interesting results about approximation by polynomials with integral coefficients, see [1,3] and the references therein. See also the other references at the end of this note). This motivates the following concept.

DISCRETIZATION AND BEST APPROXIMATION IN BESOV SPACES

In this paper we prove a theorem about existence of best approximation in a class of spaces involving Besov spaces, via a discretization technique. It is a consequence of this theorem that rational functions and exponencial sums are proximinal subsets of B∞,qa (π). It is also proved the proximinality of Rmn[a, b] in Bp,qa (π) for arbitrary p,q and a.