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.
 a. Scaling equation.
 b. Support of scaling function.
 c. Piecewise linear MRA, part 1.
 d. Orthonormal system of translates.
 e. Approximation by system of translates.
 f. Orthogonalization of system of translates.
 g. Piecewise linear MRA, part 2.
 h. Construction of MRA summary.
 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.
 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.

## Scaling equation.

roposition

(Scaling equation) Let be an MRA. There exists a sequence such that Furthermore

Proof

The equation follows from the proposition ( Subspace bases ) and the definition ( Multiresolution analysis )-1 for . The inclusion follows from the proposition ( Parseval equality ).

We take Fourier transform of : and use the formula ( Property of scale and transport 6 ):

Definition

(Scaling filter and auxiliary function) In context of the proposition ( Scaling equation ), the sequence is called "scaling filter" and the function is called "auxiliary function".

 Notation. Index. Contents.