The purpose of this tutorial is to describe the interplay between three subjects: function spaces, wavelet expansions, and multifractal analysis. Some relationships are now classical. Wavelet bases were immediately considered as remarkable by analysts because they are unconditional bases of ‘most’ function spaces. This property is a key feature of the denoising algorithms of Donoho, for instanc...

Let L be an Archimedean Riesz space with a weak order unit u. A sufficient condition under which Dedekind [σ-]completeness of the principal ideal Au can be lifted to L is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of C(X)-spaces. Similar results are obtained for the Riesz spaces Bn(T ), n = 1, 2, . . . , of all functions o...

In this paper, will be presented the concept of Borel sets and Baire on normal topological spaces locally compact as new types λ – system study basic characteristics related to each them obtain results them.

One way to formulate the Baire Category Theorem is that no compact space can be covered by countably many nowhere dense sets. Soon after Cohen’s discovery of forcing, it was realized that it was natural to consider strengthenings of this statement in which one replaces countably many with א1-many. Even taking the compact space to be the unit interval, this already implies the failure of the Con...

We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a ...