# example of construction of a Schauder basis

Consider an uniformly continuous function $f:[0,1]\rightarrow\mathbb{R}$. A Schauder basis $\{f_{n}(x)\}_{0}^{\infty}\in C[0,1]$ is constructed. For this purpose we set $f_{0}(x)=1$, $f_{1}(x)=x$. Let us consider the sequence of semi-open intervals in $[0,1]$

 $I_{n}=[2^{-k}(2n-2),2^{-k}(2n-1)),\qquad J_{n}=[2^{-k}(2n-1),2^{-k}2n),$

where $2^{k-1}, $k\geq 1$. Define now

 $\displaystyle f_{n}(x)$ $\displaystyle=$ $\displaystyle\left\{\begin{array}[]{ll}2^{k}[x-(2^{-k}(2n-2)-1)]&\text{if}\,\,% x\in I_{n},\\ 1-2^{k}[x-(2^{-k}(2n-1)-1)]&\text{if}\,\,x\in J_{n},\\ 0&\text{otherwise.}\end{array}\right.$

Geometrically these functions form a sequence of triangular functions of height one and width $2^{-(k-1)}$, sweeping $[0,1]$. So that if $f\in C([0,1])$, it is expressible in Fourier series $f(x)\sim\sum_{n=0}^{\infty}c_{n}f_{n}(x)$ and computing the coefficients $c_{n}$ by equating the values of $f(x)$ and the series at the points $x=2^{-k}m$, $m=0,1,\ldots,2^{k}$. The resulting series converges uniformly to $f(x)$ by the imposed premise.

Title example of construction of a Schauder basis ExampleOfConstructionOfASchauderBasis 2013-03-22 17:49:18 2013-03-22 17:49:18 perucho (2192) perucho (2192) 5 perucho (2192) Example msc 15A03 msc 42-00