Orthogonalization of a system of functions

The construction, for a given system of functions $\{f_n\}$ which are square integrable on the segment $[a,b]$, of an orthogonal system of functions $\{\phi_n\}$ by using a process of orthogonalization or by extending the functions $f_n$ to a larger interval $[c,d]$, $c<a<b<d$.

The use of the Schmidt orthogonalization process for a complete system of functions $\{f_n\}$ always reduces it to a complete orthonormal system $\{\phi_n\}$, and given a corresponding choice of the sequence $\{f_n\}$, permits the construction of a system which possesses some good properties. In this way, for example, the Franklin system (see Orthogonal series) is created, which is a basis in $C[0,1]$ and in $L_p[0,1]$, $p\geq1$.

Orthogonalization of a system of functions by extension to a larger interval was first introduced by I. Schur (see [1]). He proved that for the existence of a system $\{\phi_n\}$, $\phi_n(x)=f_n(x)$, $x\in[a,b]$, $0<a<b<1$, orthonormal in $L_2[0,1]$, it is necessary and sufficient that the condition

$$\sup\int\limits_a^b\left[\sum\xi_if_i(x)\right]^2dx=1$$

be fulfilled, where the supremum is taken over all $\{\xi_i\}$ with $\sum\xi_i^2=1$. Necessary and sufficient conditions have also been found such that, when these are fulfilled, one can obtain a complete orthonormal system $\{\phi_n\}$ by means of such an orthogonalization (see [2]).

A number of constructions of orthogonalization by extension of functions are given by D.E. Men'shov [3]. They are used to prove theorems on the accuracy of the condition $\sum a_n^2\ln^2n<\infty$ for the almost-everywhere convergence of an orthogonal series $\sum a_n\phi_n(x)$.

References

[1]	S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[2]	A.M. Olevskii, "On the extension of a sequence of functions to a complete orthonormal system" Math. Notes , 6 : 6 (1969) pp. 908–913 Mat. Zametki , 6 : 6 (1969) pp. 737–747
[3]	D.E. Men'shov, "Sur les séries des fonctions orthogonales bornees dans leur ensemble" Mat. Sb. , 3 (1938) pp. 103–120
[4]	Ph. Franklin, "A set of continuous orthogonal functions" Math. Ann. , 100 (1928) pp. 522–529

Comments

The Schmidt orthogonalization process is often called the Gram–Schmidt orthogonalization process.