Orthogonalization of system of translates.

roposition

(OST property 3) Assume that

1. ,

2. has compact support,

3. there exist constants such that

(Riesz basis condition)

then there exists a function $g_{0}$ such that

b. is OST,

c. . The closure is taken in $L^{2}$ .

Proof

We look for $g_{0}$ in . Then (a) is trivial and (c) would be satisfied (see below). To get (b) we use the proposition ( OST property 1 ).

We set MATH We need to achieve (a) because (see the formula ( Property of scale and transport 3 )) and we need MATH to achieve (b) (see the proposition ( OST property 1 )). We take the Fourier transform of $\left( \#\right)$ : MATH We use the formula ( Property of scale and transport 4 ): MATH Hence MATH Therefore, we would like to choose to satisfy $\left( \&\right)$ according to the rule MATH Hence, we have if we establish that the RHS is in (see the sections ( Fourier series ) and ( Fourier analysis in Hilbert space )). But this is evident from the conditions (1),(2) and (3).

It remains to prove that $\left( \#\right)$ with implies (c). The statement is trivial. We need to prove . Let so that MATH We seek such that MATH or MATH

Thus we need to have and MATH under assumptions that . Furthermore, by the condition (2), the summation in and the number of is finite. Hence, is finite dimensional linear equation. The matrix of has the form MATH for some index ranges and of the same length depending on diameter of . Since has only a finite number non-zero coordinates, the matrix is semi-diagonal. Hence, $\det C\not =0$ and always have a solution.