Function Spaces and Reconstruction

Abstract

A reconstruction problem is solved when we are able to find a function that models or describes the behavior of data. Although in each problem there will be a particular method to obtain or define this function, many of them are included in the theory of function spaces. In this chapter we will see the alphabet that permit us to understand the language for the rest of the book. The basic idea is to consider functions as simple points that behaves with the properties of Euclidean space, so we begin by generalizing the ideas of distance and orthogonality of three dimensional space to norms and inner products in function spaces. Next we study integration by parts and its application to define distributions and Sobolev spaces. Distribution theory builds a bridge between discrete and continuous processes by extending the concept of differentiability of a function.