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Difference between revisions of "Faber-Schauder system"

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A system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f0380201.png" /> on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f0380202.png" /> constructed as follows using an arbitrary countable sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f0380203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f0380204.png" />, that is everywhere dense in this interval. Set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f0380205.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f0380206.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f0380207.png" /> is linear on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f0380208.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f0380209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802011.png" />, then one divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802012.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802013.png" /> parts by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802014.png" /> and one chooses the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802016.png" />, that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802017.png" />. Then one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802019.png" />, and extends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802020.png" /> linearly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802022.png" />. Outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802023.png" /> one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802024.png" /> equal to zero.
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A system of functions $\{\phi_n(t)\}_{n=1}^\infty$ on an interval $[a,b]$ constructed as follows using an arbitrary countable sequence of points $\{w_n\}_{n=1}^\infty$, $w_1=a,w_2=b$, that is everywhere dense in this interval. Set $\phi_1(t)\equiv1$ on $[a,b]$. The function $\phi_2(t)$ is linear on $[a,b]$ such that $\phi_2(a)=0$, $\phi_2(b)=1$. If $n>2$, then one divides $[a,b]$ into $n-2$ parts by the points $w_1,\dots,w_{n-1}$ and one chooses the interval $[w_i,w_k]$, $w_1<w_k$, that contains $w_n$. Then one sets $\phi_n(w_i)=\phi_n(w_k)=0$, $\phi_n(w_n)=1$, and extends $\phi_n(t)$ linearly to $[w_i,w_n]$ and $[w_n,w_k]$. Outside $(w_i,w_k)$ one sets $\phi_n(t)$ equal to zero.
  
In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802027.png" /> is the sequence of all dyadic rational points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802028.png" />, enumerated in the natural way (that is, in the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802029.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802030.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802031.png" />), the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802032.png" /> (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802033.png" />) first appeared in the work of G. Faber [[#References|[1]]]. He considered it (with another normalization) as the system of indefinite integrals of the [[Haar system|Haar system]] supplemented by the function that is identically equal to one. In the general case, the construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802034.png" /> was carried out by J. Schauder, and so a Faber–Schauder system is also called a Schauder system.
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In the case when $a=0$, $b=1$, and $\{w_n\}$ is the sequence of all dyadic rational points in $[0,1]$, enumerated in the natural way (that is, in the order $0,1,1/2,1/4,3/4,\dots,1/2^m,3/2^m,\dots,(2^m-1)/2,\dots$), the system $\{\phi_n(t)\}$ (denoted by $\{F_n(t)\}$) first appeared in the work of G. Faber [[#References|[1]]]. He considered it (with another normalization) as the system of indefinite integrals of the [[Haar system|Haar system]] supplemented by the function that is identically equal to one. In the general case, the construction of $\{\phi_n(t)\}$ was carried out by J. Schauder, and so a Faber–Schauder system is also called a Schauder system.
  
The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802035.png" /> is a basis of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802036.png" /> of all continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802038.png" /> with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802039.png" /> (see [[#References|[1]]], [[#References|[2]]] or [[#References|[3]]]). If one applies the Schmidt orthogonalization process to the Faber system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038020/f03802041.png" />, the [[Franklin system|Franklin system]] is obtained.
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The system $\{\phi_n(t)\}$ is a basis of the space $C[a,b]$ of all continuous functions $f$ on $[a,b]$ with norm $\|f\|=\max_{a\leq t\leq b}|f(t)|$ (see [[#References|[1]]], [[#References|[2]]] or [[#References|[3]]]). If one applies the Schmidt orthogonalization process to the Faber system $\{F_n(t)\}$ on $[0,1]$, the [[Franklin system|Franklin system]] is obtained.
  
 
The Faber–Schauder system was the first example of a basis of the space of continuous functions.
 
The Faber–Schauder system was the first example of a basis of the space of continuous functions.

Latest revision as of 11:23, 6 September 2014

A system of functions $\{\phi_n(t)\}_{n=1}^\infty$ on an interval $[a,b]$ constructed as follows using an arbitrary countable sequence of points $\{w_n\}_{n=1}^\infty$, $w_1=a,w_2=b$, that is everywhere dense in this interval. Set $\phi_1(t)\equiv1$ on $[a,b]$. The function $\phi_2(t)$ is linear on $[a,b]$ such that $\phi_2(a)=0$, $\phi_2(b)=1$. If $n>2$, then one divides $[a,b]$ into $n-2$ parts by the points $w_1,\dots,w_{n-1}$ and one chooses the interval $[w_i,w_k]$, $w_1<w_k$, that contains $w_n$. Then one sets $\phi_n(w_i)=\phi_n(w_k)=0$, $\phi_n(w_n)=1$, and extends $\phi_n(t)$ linearly to $[w_i,w_n]$ and $[w_n,w_k]$. Outside $(w_i,w_k)$ one sets $\phi_n(t)$ equal to zero.

In the case when $a=0$, $b=1$, and $\{w_n\}$ is the sequence of all dyadic rational points in $[0,1]$, enumerated in the natural way (that is, in the order $0,1,1/2,1/4,3/4,\dots,1/2^m,3/2^m,\dots,(2^m-1)/2,\dots$), the system $\{\phi_n(t)\}$ (denoted by $\{F_n(t)\}$) first appeared in the work of G. Faber [1]. He considered it (with another normalization) as the system of indefinite integrals of the Haar system supplemented by the function that is identically equal to one. In the general case, the construction of $\{\phi_n(t)\}$ was carried out by J. Schauder, and so a Faber–Schauder system is also called a Schauder system.

The system $\{\phi_n(t)\}$ is a basis of the space $C[a,b]$ of all continuous functions $f$ on $[a,b]$ with norm $\|f\|=\max_{a\leq t\leq b}|f(t)|$ (see [1], [2] or [3]). If one applies the Schmidt orthogonalization process to the Faber system $\{F_n(t)\}$ on $[0,1]$, the Franklin system is obtained.

The Faber–Schauder system was the first example of a basis of the space of continuous functions.

References

[1] G. Faber, "Ueber die Orthogonalfunktionen des Herrn Haar" Jahresber. Deutsch. Math. Verein. , 19 (1910) pp. 104–112
[2] J. Schauder, "Eine Eigenschaft des Haarschen Orthogonalsystem" Math. Z. , 28 (1928) pp. 317–320
[3] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)


Comments

For the (Gram–)Schmidt orthogonalization process cf. Orthogonalization; Orthogonalization method.

References

[a1] Z. Semadeni, "Schauder bases in Banach spaces of continuous functions" , Springer (1982)
How to Cite This Entry:
Faber-Schauder system. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Faber-Schauder_system&oldid=22397
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article