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.
 a. Biorthogonal bases.
 b. Riesz bases.
 c. Generalized multiresolution analysis.
 d. Dual generalized multiresolution analysis.
 e. Dual wavelets.
 f. Orthogonality across scales.
 g. Biorthogonal QMF conditions.
 h. Vanishing moments for biorthogonal wavelets.
 i. Compactly supported smooth biorthogonal wavelets.
 j. Spline functions.
 k. Calculation of spline biorthogonal wavelets.
 l. Symmetric biorthogonal wavelets.
 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.

## Riesz bases.

efinition

(Riesz basis) A collection of functions is a Riesz basis for the (closure taken in ) iff

(a). are linearly independent,

(b). such that we have

 (Riesz frame)

Proposition

(Biorthogonality criteria 1) Let . Then is biorthogonal to iff

Proof

is similar to the proof of the proposition ( OST property 1 ).

Proposition

(Existence of biorthogonal basis 1) If has the property

 (Frame formula 1)
then there exists a biorthogonal basis with the function given by

Proof

Apply the proposition ( Biorthogonality criteria 1 ).

Proposition

(Frame property 1) If satisfies the property ( Frame formula 1 ) then for where

Proof

We calculate hence Make change and use . and we arrive to the desired conclusion.

Proposition

(Frame property 2)

1. If a function satisfies the formula ( Frame formula 1 ) then there exist such that we have

 (Riesz property)

2. If a function has compact support and satisfies the formula ( Riesz property ) then it satisfies the formula ( Frame formula 1 ).

Proof

(1). For a function we calculate We use the formula ( Property of scale and transport 4 ). We continue calculation of We make a change . The function has period 1. The function has period 1 and the above expression is the Fourier coefficient of it. By proposition ( Parseval equality ) we have We substitute the formula ( Frame formula 1 ): so that and use the following consequence of the proposition ( Frame property 1 ): thus

Proof

(2) According to the proposition ( Property of transport 2 ), the function takes the form for some finite sequence of real numbers . For this reason we already have To prove the part it suffices to disprove existence of such that . This is so because is periodic.

For to vanish, every term has to vanish. In other words, we aim to show that there cannot exist a such that or

In the part (1) of the proof we obtained the generic relationship . We will reach our goal if we can find a sequence : where the is Fourier transform of : Indeed, if such exists then, according to , and we would have a contradiction with the formula ( Riesz property ).

Existence of such is standard. We approximate on by an -indexed sequence of periodic functions, obtain its Fourier coefficients and use them in the combination .

Proposition

(Frame property 3) If functions satisfy the conditions

(a) is compactly supported,

(b) is a Riesz basis for ,

(c) is biorthogonal to .

then

(1)

(2) there exist such that

Proof

It suffices to prove (1) and (2) for . Extending to the closure is a standard exercise because all involved operations are -continuous.

Let then by applying we get hence (1).

According to the proposition ( Biorthogonality criteria 1 ), and by condition (b) and proposition ( Frame property 2 )-2, for some From we derive for some .

We now apply the proposition ( Frame property 1 ): By the proposition ( Parseval equality ), hence (2).

 Notation. Index. Contents.