Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.
 J. Construction of (G)MRA and wavelets on an interval.
 a. Adapting MRA to the interval [0,1].
 b. Adapting wavelets to interval [0,1].
 c. Adapting GMRA to interval [0,1].
 d. Adapting dual wavelets to interval [0,1].
 e. Constructing dual GMRA on [0,1] with boundary conditions.
 3 Finite element method.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Adapting dual wavelets to interval [0,1].

e assume that the conditions ( Biorthogonal scaling functions ) and ( Sufficiently fine scale 2 ) hold and build on results of the previous section. We assume that and .

We introduce the spaces and via the relationships

The bases for and are constructed as follows. Let The space has dimension . Hence, we construct a part of the basis for by taking and . The remaining functions from are derived by constructing the functions

 (Wavelets on 01 step 1)
where Similarly, on the right hand side of the interval we construct the same and by setting We form the linear combinations for some finite sequences , , , determined below. The wavelet dual bases of the space take the form The , , , are chosen to satisfy

Let for some matrixes and columns Then

We conclude as in the previous section:

Remark

(Dimension mismatch) Note that we generally have This means that we get at least 2 functions too many when constructing basis for the space . As a result, when we perform the biorthogonalization of boundary functions, at least two of the resulting functions (one for every boundary) vanish, see the proposition ( Linear independence from biorthogonality ). We discard such functions and end up with a basis of correct dimensionality.

 Notation. Index. Contents.