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 $2^{k-1}<n\leq 2^{k}$ , $k\geq 1$ . 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
Canonical name	ExampleOfConstructionOfASchauderBasis
Date of creation	2013-03-22 17:49:18
Last modified on	2013-03-22 17:49:18
Owner	perucho (2192)
Last modified by	perucho (2192)
Numerical id	5
Author	perucho (2192)
Entry type	Example
Classification	msc 15A03
Classification	msc 42-00