I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 1 Calculational Linear Algebra.
 2 Wavelet Analysis.
 A. Elementary definitions of wavelet analysis.
 B. Haar functions.
 C. Multiresolution analysis.
 D. Orthonormal wavelet bases.
 E. Discrete wavelet transform.
 F. Construction of MRA from scaling filter or auxiliary function.
 a. Quadrature mirror filter (QMF) conditions.
 b. Recovering scaling function from auxiliary function. Cascade algorithm.
 c. Recovering MRA from auxiliary function.
 G. Consequences and conditions for vanishing moments of wavelets.
 H. Existence of smooth compactly supported wavelets. Daubechies polynomials.
 I. Semi-orthogonal wavelet bases.
 J. Construction of (G)MRA and wavelets on an interval.
 3 Finite element method.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Recovering MRA from auxiliary function.

roposition

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

1. is given by a finite filter :

2. satisfies the conditions ( QMF conditions ).

Then there exists an MRA such that the recovered from via is a scaling function of .

Proof

We form 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 we conclude We would like to take inverse Fourier transform of .

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

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

The set of functions with compact support constitutes a dense set in and for each function on such dense set we have Hence, the above convergence result extends to all by the following standard argument. Let and have compact support and , as . 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.

 Notation. Index. Contents.