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.

## Piecewise linear MRA, part 2.

e continue the construction of the section ( Piecewise linear MRA ). It remains to satisfy the requirement ( Multiresolution analysis )-5. We use the proposition ( OST property 3 ).

In context of the proposition ( OST property 3 ), let

 (Starting scaling function)

Proposition

(V0 for linear MRA) Given the formulas ( V for linear MRA 0 ) and ( Starting scaling function ) we have

Proof

Direct verification.

It remains to verify that of the formula ( Starting scaling function ) satisfies the formula ( Riesz basis condition ).

We could evaluate and then estimate as in the formula ( Riesz basis condition ). However there is a better way for with finite support.

Proposition

(Shifted Fourier transform equality) If then

Proof

The statement is a direct consequence of the proposition ( Property of transport 1 ).

For the function of the form ( Starting scaling function ) only three terms are non-zero: Hence The property ( Riesz basis condition ) is now evident and existence of the function of the definition ( Multiresolution analysis )-5 follows from the proposition ( OST property 3 ).

 Notation. Index. Contents.