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example of construction of a Schauder basis
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(Example)
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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 $[0,1]$ \begin{equation*} I_n=[2^{-k}(2n-2),2^{-k}(2n-1)), \qquad J_n=[2^{-k}(2n-1),2^{-k}2n), \end{equation*}where $2^{k-1}<n\leq 2^k$ , $k\geq
1$ . Define now
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 $f(x)\sim \sum_{n=0}^\infty c_nf_n(x)$ 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.
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"example of construction of a Schauder basis" is owned by perucho.
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Cross-references: premise, converges uniformly, points, series, coefficients, Fourier series, expressible, width, height, functions, intervals, sequence, Schauder basis, uniformly continuous function
This is version 2 of example of construction of a Schauder basis, born on 2008-02-18, modified 2008-02-18.
Object id is 10285, canonical name is ExampleOfConstructionOfASchauderBasis.
Accessed 640 times total.
Classification:
| AMS MSC: | 42-00 (Fourier analysis :: General reference works ) | | | 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank) |
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Pending Errata and Addenda
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![$\displaystyle \left\{\begin{array}{ll} 2^k[x-(2^{-k}(2n-2)-1)] & \text{if}\,\, ... ...n-1)-1)] & \text{if}\,\, x\in J_n, \ 0 & \text{otherwise.} \end{array}\right.$](http://images.planetmath.org:8080/cache/objects/10285/js/img3.png "$\displaystyle \left\{\begin{array}{ll} 2^k[x-(2^{-k}(2n-2)-1)] & \text{if}\,\, ... ...n-1)-1)] & \text{if}\,\, x\in J_n, \ 0 & \text{otherwise.} \end{array}\right.$")