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.
 G. Consequences and conditions for vanishing moments of wavelets.
 H. Existence of smooth compactly supported wavelets. Daubechies polynomials.
 I. Semi-orthogonal wavelet bases.
 a. Biorthogonal bases.
 b. Riesz bases.
 c. Generalized multiresolution analysis.
 d. Dual generalized multiresolution analysis.
 e. Dual wavelets.
 f. Orthogonality across scales.
 g. Biorthogonal QMF conditions.
 h. Vanishing moments for biorthogonal wavelets.
 i. Compactly supported smooth biorthogonal wavelets.
 j. Spline functions.
 k. Calculation of spline biorthogonal wavelets.
 l. Symmetric biorthogonal wavelets.
 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.

## Compactly supported smooth biorthogonal wavelets. otivation and notation of this section is based on the section ( Smooth compactly supported wavelets ). We calculate biorthogonal wavelets bases , such that both and have compact support. However, we lose the property ( Existence of wavelets with orthogonality across scales )-d. In particular, we no longer have neither nor .

Proposition

(Existence of biorthogonal compactly supported wavelets ) There exist smooth compactly supported real valued biorthogonal wavelets calculated according to the following procedure:

1. Pick a number . It will influence the length of the scaling filters and the size of support of , see the proposition ( Support of scaling function ).

2. Set where the are binomial coefficients, .

4. Find any pair of finite sequences such that 5. Use the proposition ( Cascade algorithm ) to find .

6. Use the proposition ( Scaling equation 2 ) to find , .

Remark

There is more then one way to achieve a match of the condition 4. One way to do it is to select a scaling function as in the proposition ( Existence of biorthogonal basis 1 ) (using the proposition ( Shifted Fourier transform equality ) to verify applicability), calculate the corresponding as in the proposition ( Scaling equation 4 ) and then seek to satisfy the condition 4. The following sections contain examples.

Proof

According to the propositions ( Sufficient conditions for vanishing moments 2 ), ( Vanishing moments vs decay at infinity ), ( Biorthogonal QMF property 1 ) we are looking for a pair of functions where the sequences are real and finite. We seek to satisfy the following conditions:    Note that We repeat calculations of the proof of the proposition ( Existence of smooth compactly supported wavelets ) and arrive to for some polynomial that has to satisfy Such polynomial was constructed in the proof of the proposition ( Existence of smooth compactly supported wavelets ).

 Notation. Index. Contents.