Faber-Schauder system

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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)