Preliminaries

We begin by recalling some basic notions of functional analysis. A measurable function $f$ belongs to the Lebesgue space $L_p\left(\mathrm{I}\mathrm{R}\right),1\le p<\infty$ if

A Hilbert space is a space where an inner product is defined. In particular, the space $L_2\left(\mathrm{I}\mathrm{R}\right)$ is a Hilbert space, where the inner product of 2 functions $f$ and $g$ is defined as:

In this presentation we work with functions defined on $\mathrm{I}\mathrm{R},$ but that take values in $C[0.1ex]0.05em1.25ex.$ Hence $\overline{g\left(x\right)}$ denotes the complex conjugate of $g\left(x\right).$ We say that 2 functions are orthogonal if their inner product is zero. A function is Hölder continuous of order $\alpha ,(0<\alpha \le 1)$ at point $x$ if :