Advanced Courses of Mathematical Analysis IV

The following sections are included:

• PREFACE

• ORGANIZING COMMITTEES

• CONTENTS

The following sections are included:

• The person

• The professor

• The researcher

• References

The following sections are included:

• Introduction

• Preliminaries

• Browder theorems

• Weyl type theorems

• Weyl type theorems for left and right polaroid operators

• Some applications

• Polaroid operators under quasi-affinity

• References

The following sections are included:

• Introduction

• Bounded, additive measures

• The dual of

• The dual of C(K), K a compact, Hausdorff space

• Finitely Additive Measures and Geometry

• ba and Banach Spaces

• References

Bandlimited functions, i.e square integrable functions on ℝ^d, d ∈ ℕ, whose Fourier transforms have bounded support, are widely used to represent signals. One problem which arises, is to find stable recovery formulae, based on evaluations of these functions at given sample points. We start with the case of equally distributed sampling points and present a method of Daubechies and DeVore to approximate bandlimited functions by quantized data.

In the case that the sampling points are not equally distributed this method will fail. We are suggesting to provide a solution to this problem in the case of scattered sample points by first approximating bandlimited functions using linear combinations of shifted Gaussians. In order to be able to do so we prove the following interpolation result.

Let (x_j : j ∈ ℤ) ⊂ ℝ be a Rieszbasis sequence. For λ > 0 and f ∈ PW, the space of square-integrable functions on ℝ, whose Fourier transforms vanish outside of [−1, 1], there is a unique sequence (a_j) ∈ ℓ₂(ℤ), so that the function

The following sections are included:

• Introduction

• Question 1

• Question 4 and new questions

• Questions 2c and 1 again

• Question 2d for a = 1

• Questions 3 and 1b

• Question 2

• A final question

• References

We give a survey of results dealing with the Banach algebra structure of the set of uniformly continuous and symmetric holomorphic functions on the ball of some classical Banach spaces.

We will review a series of results of Prof. Antonio Aizpuru on the local theory of normed spaces since 2002. However, we will begin by introducing 1997 Aizpuru's E-Property.

This survey is articulated around two major axis. The first one concerns the Kaplansky problem; the history of the problem and several results are presented. The second one concerns some new preserver problems (concerning the generalized inverse, Fredholm or semi-Fredholm operators). The common point of these results is that they are interesting only in the infinite dimensional situation. Several open questions are mentioned over all the paper.

We survey some connections of the bounded approximation property with Banach operator ideals, both historical and recent ones. The exposition is self-contained and includes an introduction to Banach operator ideals.

The following sections are included:

• Introduction

• Continuous functions

• Lipschitz functions

• References

This survey note is dedicated to the memory of Prof. Antonio Aizpuru Tomás (Valencia, 1954 - Cádiz, 2008). We present an overview of some of Prof. Aizpuru's results on the applications of lineability to Series and Summability Methods. We will go through his last papers and some of his results relating these two concepts.

We survey some recent works ([3,11,13]) concerning the study of the norm of the Hardy operator minus the Identity. In particular, optimal estimates are found for the cone of radially decreasing functions on the minimal Lorentz spaces Λ(X) (restricted type estimates), and a new Banach function space, closely related to Λ(X), is also naturally defined. We also mention a recent solution to a conjecture of Kruglyak and Setterqvist.

The following sections are included:

• AUTHOR INDEX