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.
 a. Vanishing moments vs decay at infinity.
 b. Vanishing moments vs approximation.
 c. Sufficient conditions for vanishing moments.
 d. Reproduction of polynomials.
 e. Smoothness of compactly supported wavelets with vanishing moments.
 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.

## Reproduction of polynomials.

roposition

(Reproduction of polynomials 1) Let be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. is compactly supported and (therefore) is finite.

2. , .

Then

Proof

According to the proposition ( Scaling equation ), thus Let be odd, then We apply the proposition ( Sufficient conditions for vanishing moments )-b

For even we have , for some integer and an odd integer . We use the proposition ( Scaling equation ) times: thus and we repeat the calculation of the odd case:

Remark

Note that is a 1-periodic function. Hence, we may consider the Fourier coefficients of : We make a change . Under conditions of the last proposition, . Thus

Proposition

(Reproduction of polynomials 2)Let be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. is compactly supported and (therefore) is finite,

2. , .

Then for .

Proof

The function has period 1. We evaluate an -th Fourier coefficient: We apply the proposition ( Reproduction of polynomials 1 ). Thus, by section ( Fourier series section ),

Proposition

(Reproduction of polynomials 3) Let be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. is compactly supported and (therefore) is finite,

2. , .

Then there exist polynomials of degree such that for .

Proof

We introduce the convenience notation then the proposition ( Reproduction of polynomials 2 ) provides For we have For we have so that For we have We use results of previous steps. We continue similarly for all up to .

Proposition

(Reproduction of polynomials 4) Let be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. is compactly supported and (therefore) is finite,

2. , .

Then there are numbers such that for .

Proof

Take linear combination of equations from the proposition ( Reproduction of polynomials 3 ).

 Notation. Index. Contents.