Editing Haar wavelet (section)

== Haar system on the unit interval and related systems ==
In this section, the discussion is restricted to the [[unit interval]] [0,&nbsp;1] and to the Haar functions that are supported on [0,&nbsp;1]. The system of functions considered by Haar in 1910,<ref>p.&nbsp;361 in {{harvtxt|Haar|1910}}</ref>
called the '''Haar system on [0,&nbsp;1]''' in this article, consists of the subset of Haar wavelets defined as
:<math>\{ t \in [0, 1] \mapsto \psi_{n,k}(t) \; : \; n, k \in \N \cup \{0\}, \; 0 \leq k < 2^n\},</math>
with the addition of the constant function '''1''' on [0,&nbsp;1].

In [[Hilbert space]] terms, this Haar system on [0,&nbsp;1] is a complete orthonormal system, ''i.e.'', an [[orthonormal basis]], for the space ''L''<sup>2</sup>([0,&nbsp;1]) of square integrable functions on the unit interval.

The Haar system on [0,&nbsp;1] &mdash;with the constant function '''1''' as first element, followed with the Haar functions ordered according to the [[Lexicographical order|lexicographic]] ordering of couples {{nowrap|(''n'', ''k'')}}&mdash; is further a [[Schauder basis#Properties|monotone]] [[Schauder basis]] for the space [[Lp space|''L''<sup>''p''</sup>([0,&nbsp;1])]] when {{nowrap|1 ≤ ''p'' &lt; ∞}}.<ref name="L. Tzafriri, 1977">see p.&nbsp;3 in [[Joram Lindenstrauss|J. Lindenstrauss]], L. Tzafriri, (1977), "Classical Banach Spaces I, Sequence Spaces", Ergebnisse der Mathematik und ihrer Grenzgebiete '''92''', Berlin: Springer-Verlag, {{ISBN|3-540-08072-4}}.</ref> 
This basis is [[Schauder basis#Unconditionality|unconditional]] when {{nowrap|1 &lt; ''p'' &lt; ∞}}.<ref>The result is due to [[Raymond Paley|R. E. Paley]], ''A remarkable series of orthogonal functions (I)'', Proc. London Math. Soc. '''34''' (1931) pp. 241-264. See also p.&nbsp;155 in J. Lindenstrauss, L. Tzafriri, (1979), "Classical Banach spaces II, Function spaces". Ergebnisse der Mathematik und ihrer Grenzgebiete '''97''', Berlin: Springer-Verlag, {{ISBN|3-540-08888-1}}.</ref>

There is a related [[Rademacher system]] consisting of sums of Haar functions, 
:<math>r_n(t) = 2^{-n/2} \sum_{k=0}^{2^n - 1} \psi_{n, k}(t), \quad t \in [0, 1], \ n \ge 0.</math>
Notice that |''r''<sub>''n''</sub>(''t'')|&nbsp;= 1 on [0,&nbsp;1). This is an orthonormal system but it is not complete.<ref>{{SpringerEOM |title=Orthogonal system}}</ref><ref>{{cite book |first1=Gilbert G. |last1=Walter |first2=Xiaoping |last2=Shen |title=Wavelets and Other Orthogonal Systems |year=2001 |location=Boca Raton |publisher=Chapman |isbn=1-58488-227-1 }}</ref>
In the language of [[probability theory]], the Rademacher sequence is an instance of a sequence of [[Independence (probability theory)|independent]] [[Bernoulli distribution|Bernoulli]] [[random variables]] with [[mean]]&nbsp;0. The [[Khintchine inequality]] expresses the fact that in all the spaces ''L''<sup>''p''</sup>([0,&nbsp;1]), {{nowrap|1 ≤ ''p'' &lt; ∞}}, the Rademacher sequence is [[Schauder basis#Definitions|equivalent]] to the unit vector basis in ℓ<sup>''2''</sup>.<ref>see for example p.&nbsp;66 in [[Joram Lindenstrauss|J. Lindenstrauss]], L. Tzafriri, (1977), "Classical Banach Spaces I, Sequence Spaces", Ergebnisse der Mathematik und ihrer Grenzgebiete '''92''', Berlin: Springer-Verlag, {{ISBN|3-540-08072-4}}.</ref> In particular, the [[Linear span#Closed linear span|closed linear span]] of the Rademacher sequence in ''L''<sup>''p''</sup>([0,&nbsp;1]), {{nowrap|1 ≤ ''p'' &lt; ∞}}, is [[Isomorphic normed spaces|isomorphic]] to ℓ<sup>''2''</sup>.

=== 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''.

=== 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.