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 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.

## Construction of MRA summary.

his section summarizes the derivations so far. This is not the only recipe for construction of an MRA. See the proposition ( Existence of smooth compactly supported wavelets ) for a more definitive procedure.

Start from a function with compact support.

Form , closure is in .

The verification of ( Multiresolution analysis )-1 should be direct.

For verification of ( Multiresolution analysis )-2 use the technique of the proposition ( Main property of Haar functions 1 ) or technique of the section ( Piecewise linear MRA ).

For verification of ( Multiresolution analysis )-3 use the technique illustrated in the section ( Piecewise linear MRA section ).

The property ( Multiresolution analysis )-4 is automatic.

Verify the condition ( Riesz basis condition ) using the proposition ( Shifted Fourier transform equality ) and achieve the orthogonality part of ( Multiresolution analysis )-5 by switching according to the recipe of the proposition ( OST property 3 ).

 Notation. Index. Contents.