example of construction of a Schauder basis

Consider an uniformly continuous function f:[0,1]. A Schauder basis {fn(x)}0C[0,1] is constructed. For this purpose we set f0(x)=1, f1(x)=x. Let us consider the sequence of semi-open intervals in [0,1]


where 2k-1<n2k, k1. Define now

fn(x) = {2k[x-(2-k(2n-2)-1)]ifxIn,1-2k[x-(2-k(2n-1)-1)]ifxJn,0otherwise.

Geometrically these functions form a sequence of triangular functions of height one and width 2-(k-1), sweeping [0,1]. So that if fC([0,1]), it is expressible in Fourier series f(x)n=0cnfn(x) and computing the coefficients cn by equating the values of f(x) and the series at the points x=2-km, m=0,1,,2k. 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