The concept of multi-resolution analysis provides families of orthogonal bases that can be generated from a common seminal function. Its main application has been in the theoretical development of the so-called wavelet functions. Here, we are going to apply multi-resolution schemes in two different settings.

The first problem deals with wavelet-based procedures for estimating functions. Estimators for probability density and regression functions are proposed, some asymptotic properties are established, and application to data sets is performed.

The other problem concerns boundary detection in some Gaussian random fields and asymptotic properties of these fields under discontinuity. The spirit of multi-resolution is presented here through the increasing resolution of the fields.

Convergence of some Gaussian random fields with discontinuities when the resolution increases is established. Justification for certain boundary detection procedures is also provided.