Editing Haar wavelet (section)

=== The Faber&ndash;Schauder system ===
The '''Faber&ndash;Schauder system'''<ref name="Faber">Faber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", ''Deutsche Math.-Ver'' (in German) '''19''': 104&ndash;112. {{issn|0012-0456}}; 
http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553</ref><ref>Schauder, Juliusz (1928), "Eine Eigenschaft des Haarschen Orthogonalsystems", ''Mathematische Zeitschrift'' '''28''': 317&ndash;320.</ref><ref>{{eom|id=f/f038020
 |title=Faber–Schauder system|first=B.I.|last= Golubov}}</ref> 
is the family of continuous functions on [0,&nbsp;1] consisting of the constant function&nbsp;'''1''', and of multiples of [[Antiderivative|indefinite integrals]] of the functions in the Haar system on&nbsp;[0,&nbsp;1], chosen to have norm&nbsp;1 in the [[Uniform norm|maximum norm]]. This system begins with ''s''<sub>0</sub>&nbsp;=&nbsp;'''1''', then {{nowrap| ''s''<sub>1</sub>(''t'') {{=}} ''t''}} is the indefinite integral vanishing at&nbsp;0 of the function&nbsp;'''1''', first element of the Haar system on [0,&nbsp;1]. Next, for every integer {{nowrap|''n'' ≥ 0}}, functions {{nowrap| ''s''<sub>''n'',''k''</sub>}} are defined by the formula
:<math>
 s_{n, k}(t) = 2^{1 + n/2} \int_0^t \psi_{n, k}(u) \, d u, \quad t \in [0, 1], \ 0 \le k < 2^n.</math>
These functions {{nowrap| ''s''<sub>''n'',''k''</sub>}} are continuous, [[Piecewise linear function|piecewise linear]], supported by the interval {{nowrap| ''I''<sub>''n'',''k''</sub>}} that also supports {{nowrap| ψ<sub>''n'',''k''</sub>}}. The function {{nowrap| ''s''<sub>''n'',''k''</sub>}} is equal to&nbsp;1 at the midpoint {{nowrap| ''x''<sub>''n'',''k''</sub>}} of the interval&nbsp;{{nowrap| ''I''<sub>''n'',''k''</sub>}}, linear on both halves of that interval. It takes values between&nbsp;0 and&nbsp;1 everywhere.

The Faber&ndash;Schauder system is a [[Schauder basis]] for the space ''C''([0,&nbsp;1]) of continuous functions on [0,&nbsp;1].<ref name="L. Tzafriri, 1977"/> 
For every&nbsp;''f'' in ''C''([0,&nbsp;1]), the partial sum
:<math> f_{n+1} = a_0 s_0 + a_1 s_1 + \sum_{m = 0}^{n-1} \Bigl( \sum_{k=0}^{2^m - 1} a_{m,k} s_{m, k} \Bigr) \in C([0, 1])</math>
of the [[series expansion]] of ''f'' in the Faber&ndash;Schauder system is the continuous piecewise linear function that agrees with&nbsp;''f'' at the {{nowrap|2<sup>''n''</sup> + 1}} points {{nowrap|''k''2<sup>&minus;''n''</sup>}}, where {{nowrap| 0 ≤ ''k'' ≤ 2<sup>''n''</sup>}}. Next, the formula
:<math> f_{n+2} - f_{n+1} = \sum_{k=0}^{2^n - 1} \bigl( f(x_{n,k}) - f_{n+1}(x_{n, k}) \bigr) s_{n, k} = \sum_{k=0}^{2^n - 1} a_{n, k} s_{n, k} </math>
gives a way to compute the expansion of ''f'' step by step. Since ''f'' is [[Heine–Borel theorem|uniformly continuous]], the sequence {''f''<sub>''n''</sub>} converges uniformly to ''f''. It follows that the Faber&ndash;Schauder series expansion of ''f'' converges in ''C''([0,&nbsp;1]), and the sum of this series is equal to&nbsp;''f''.