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 1.

n this section we start construction of a piecewise linear MRA.

Let

 (V for linear MRA 0)
We satisfy the requirement ( Multiresolution analysis )-4 by taking
 (V for linear MRA d)

The requirement ( Multiresolution analysis )-3 is satisfied because is allowed to approach . Hence, we must have and linearity on and for any . Thus, for any such .

The requirement ( Multiresolution analysis )-2 is satisfied because one can -approximate a function with a sequence of linear functions on for growing .

The requirement ( Multiresolution analysis )-1 is trivial.

The requirement ( Multiresolution analysis )-5 is the key non-trivial component. It requires some preliminary development.

 Notation. Index. Contents.