Recovering MRA from auxiliary function.

roposition

(Recovering MRA from auxiliary function 1) Suppose a function satisfies the following conditions:

1. $m_{0}$ is given by a finite filter : MATH

2. $m_{0}$ satisfies the conditions ( QMF conditions ).

Then there exists an MRA such that the $\phi$ recovered from $m_{0}$ via MATH is a scaling function of .

Proof

We form MATH Then the condition ( Multiresolution analysis )-4 and are evident. The orthogonality part of ( Multiresolution analysis )-5 follows from the QMF conditions, see the proof of the proposition ( Scaling equation 3 ).

We now prove the condition ( Multiresolution analysis )-1.

From the formula $\left( \#\right)$ we conclude MATH We would like to take inverse Fourier transform of $\left( @\right)$ .

By condition 1 and according to the section ( Fourier transform of delta function ), MATH For a general function , MATH thus MATH We now take the inverse Fourier transform of the equality $\left( @\right)$ . The product becomes convolution: MATH Thus we have the condition ( Multiresolution analysis )-1: MATH

We now verify the condition ( Multiresolution analysis )-3. It suffices to show that we have MATH First, we consider function with compact support: MATH We estimate MATH MATH MATH MATH Thus MATH

The set of functions with compact support constitutes a dense set in $L^{2}$ and for each function on such dense set we have MATH Hence, the above convergence result extends to all by the following standard argument. Let $g\in L^{2}$ and have compact support and , as $k\rightarrow\infty$ . MATH If does not converge to zero then there is a subsequence such that is separated from zero. But then we arrive to contradiction because everything on the RHS of can be arbitrarily small.

The verification of condition ( Multiresolution analysis )-2 may be found in [Mallat] , page 276.