On Generalized Hölder-Zygmund Spaces

To study the regularity of functions, many functional spaces have been introduced during the 20th century. Among them, let us mention the Hölder-Zygmund spaces \Lambda^{\alpha}(\R^{d}) and the Besov spaces B_{p,q}^{\alpha}(\R^{d}) where \alpha>0 somehow indicates the regularity of their elements (p,q\in ]0,+\infty]). The Hölder-Zygmund spaces are particular cases of Besov spaces in the sense that \Lambda^{\alpha}(\R^{d})=B_{\infty,\infty}^{\alpha}(\R^{d}).

A generalization of Besov spaces has been introduced in the middle of the seventies and is still studied nowadays. This new type of space allows a deepest study of the regularity of functions. In this thesis, we start from this generalization in order to introduce a generalization of Hölder-Zygmund spaces.

The first aim of this thesis is to show that most classical properties of Hölder-Zygmund spaces can be transposed to their generalized version. Among others, a complete characterization of these spaces in terms of wavelet coefficients is proved, which opens their use in the context of the signal analysis.

The second aim of this thesis is to introduce a generalized version of the pointwise Hölder spaces similarly to their global version. We then show that most properties of the global spaces can be transposed to their generalized pointwise version.

Finally, we study the regularity of some financial stochastic processes.