Faber-Schauder system

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)