Faber-Schauder system
A system of functions on an interval
constructed as follows using an arbitrary countable sequence of points
,
, that is everywhere dense in this interval. Set
on
. The function
is linear on
such that
,
. If
, then one divides
into
parts by the points
and one chooses the interval
,
, that contains
. Then one sets
,
, and extends
linearly to
and
. Outside
one sets
equal to zero.
In the case when ,
, and
is the sequence of all dyadic rational points in
, enumerated in the natural way (that is, in the order
), the system
(denoted by
) 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
was carried out by J. Schauder, and so a Faber–Schauder system is also called a Schauder system.
The system is a basis of the space
of all continuous functions
on
with norm
(see [1], [2] or [3]). If one applies the Schmidt orthogonalization process to the Faber system
on
, 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) |
Faber-Schauder system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Faber-Schauder_system&oldid=33251