Editing Haar wavelet (section)

=== The Franklin system ===
The '''Franklin system''' is obtained from the Faber&ndash;Schauder system by the [[Gram–Schmidt process|Gram&ndash;Schmidt orthonormalization procedure]].<ref>see Z. Ciesielski, ''Properties of the orthonormal Franklin system''. Studia Math. 23 1963 141–157.</ref><ref>Franklin system. B.I. Golubov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Franklin_system&oldid=16655</ref>
Since the Franklin system has the same [[linear span]] as that of the Faber&ndash;Schauder system, this span is dense in ''C''([0,&nbsp;1]), hence in ''L''<sup>2</sup>([0,&nbsp;1]). The Franklin system is therefore an orthonormal basis for ''L''<sup>2</sup>([0,&nbsp;1]), consisting of continuous piecewise linear functions. P. Franklin proved in 1928 that this system is a Schauder basis for ''C''([0,&nbsp;1]).<ref>Philip Franklin, ''A set of continuous orthogonal functions'', Math. Ann. 100 (1928), 522-529. {{doi|10.1007/BF01448860}}</ref> 
The Franklin system is also an unconditional Schauder basis for the space ''L''<sup>''p''</sup>([0,&nbsp;1]) when {{nowrap|1 &lt; ''p'' &lt; ∞}}.<ref name=Bo>S. V. Bočkarev, ''Existence of a basis in the space of functions analytic in the disc, and some properties of Franklin's system''. Mat. Sb. '''95''' (1974), 3–18 (Russian). Translated in Math. USSR-Sb. '''24''' (1974), 1–16.</ref>
The Franklin system provides a Schauder basis in the [[disk algebra]] ''A''(''D'').<ref name=Bo />
This was proved in 1974 by Bočkarev, after the existence of a basis for the disk algebra had remained open for more than forty years.<ref>The question appears p.&nbsp;238, §3 in Banach's book, {{citation|first=Stefan|last=Banach|author-link=Stefan Banach|url=http://matwbn.icm.edu.pl/kstresc.php?tom=1&wyd=10|title=Théorie des opérations linéaires|publication-place=Warszawa|publisher=Subwencji Funduszu Kultury Narodowej|year=1932|series=Monografie Matematyczne|volume=1|zbl=0005.20901}}.  The disk algebra ''A''(''D'') appears as Example&nbsp;10, p.&nbsp;12 in Banach's book.</ref>

Bočkarev's construction of a Schauder basis in ''A''(''D'') goes as follows: let&nbsp;''f'' be a complex valued [[Lipschitz continuity|Lipschitz function]] on [0,&nbsp;π]; then&nbsp;''f'' is the sum of a [[Fourier series|cosine series]] with [[Absolute convergence|absolutely summable]] coefficients. Let&nbsp;''T''(''f'') be the element of ''A''(''D'') defined by the complex [[power series]] with the same coefficients,

:<math> \left\{ f : x \in [0, \pi] \rightarrow \sum_{n=0}^\infty a_n \cos(n x) \right\} \longrightarrow \left\{ T(f) : z \rightarrow \sum_{n=0}^\infty a_n z^n, \quad |z| \le 1 \right\}.</math>
Bočkarev's basis for ''A''(''D'') is formed by the images under&nbsp;''T'' of the functions in the Franklin system on&nbsp;[0,&nbsp;π]. Bočkarev's equivalent description for the mapping&nbsp;''T'' starts by extending ''f'' to an [[Even and odd functions|even]] Lipschitz function&nbsp;''g''<sub>1</sub> on [&minus;π,&nbsp;π], identified with a Lipschitz function on the [[unit circle]]&nbsp;'''T'''. Next, let ''g''<sub>2</sub> be the [[Hardy space conjugate function|conjugate function]] of&nbsp;''g''<sub>1</sub>, and define ''T''(''f'') to be the function in&nbsp;''A''(''D'') whose value on the boundary '''T''' of&nbsp;''D'' is equal to&nbsp;{{nowrap|''g''<sub>1</sub> + i''g''<sub>2</sub>}}.

When dealing with 1-periodic continuous functions, or rather with continuous functions ''f'' on [0,&nbsp;1] such that {{nowrap|''f''(0) {{=}} ''f''(1)}}, one removes the function {{nowrap| ''s''<sub>1</sub>(''t'') {{=}} ''t''}} from the Faber&ndash;Schauder system, in order to obtain the '''periodic Faber&ndash;Schauder system'''. The '''periodic Franklin system''' is obtained by orthonormalization from the periodic Faber&ndash;-Schauder system.<ref name="Prz">See p.&nbsp;161, III.D.20 and p.&nbsp;192, III.E.17 in {{citation
 | last=Wojtaszczyk | first= Przemysław
 | title = Banach spaces for analysts
 | series = Cambridge Studies in Advanced Mathematics
 | volume = 25
 | publisher = Cambridge University Press 
 | location = Cambridge
 | year= 1991
 | pages = xiv+382
 | isbn = 0-521-35618-0 
}}</ref>
One can prove Bočkarev's result on ''A''(''D'') by proving that the periodic Franklin system on [0,&nbsp;2π] is a basis for a Banach space ''A''<sub>''r''</sub> isomorphic to ''A''(''D'').<ref name="Prz" /> 
The space ''A''<sub>''r''</sub> consists of complex continuous functions on the unit circle '''T''' whose [[Harmonic conjugate|conjugate function]] is also continuous.