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.
 3 Finite element method.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Existence of smooth compactly supported wavelets. Daubechies polynomials.

roposition

(Existence of smooth compactly supported wavelets) There exist smooth compactly supported wavelets calculated according to the following procedure:

1. Pick a number . It will determine the size of support and the length of the scaling filter and the size of support of , see the proposition ( Support of scaling function ).

2. Set where the are binomial coefficients, .

3. Set for some numbers defined in the next step.

4. Match to find .

5. Set

6. Use the proposition ( Cascade algorithm ) to find .

7. Use the proposition ( Scaling equation 2 ) to find .

Remark

The step 4 is possible for the following reasons. We have for some numbers and the sum is real-valued for . Therefore the has to be real-valued by orthogonality of . We calculate and the imaginary terms must vanish. Hence, is a polynomial of of degree and we can always do the match.

Remark

It follows from the step 5 and calculations of the previous remark that is of the form Thus and the length of is , see the proposition ( Support of scaling function ).

Remark

Nothing prevents us from taking summation ranges instead of . The results would be the same up to a shift.

Proof

of existence. The key motivational statement is the proposition ( Recovering MRA from auxilliary function 1 ). According to the propositions ( Sufficient conditions for vanishing moments ), ( Vanishing moments vs decay at infinity ), ( QMF property 1 ) we are looking for a function where the sequence is finite due to compactness of support and real because we seek to build an MRA with real valued and thus have the proposition ( Scaling equation ). The is sought to satisfy the following conditions: for some finite sequence of numbers .

We have

Let We must have so that , . Hence for some finite set . Also Hence, the requirement takes the form for some polynomial . Therefore, we are looking for that satisfies We find such by performing the following calculation: We make change in the second sum, ,

Remark

on smoothness. According to the proposition ( Reproduction of polynomials 4 ), there exist numbers s.t. Also,

Thus, the number of discontinuities has to be finite.

Consider the case . The sum replicates a constant. . . Thus, can only jump at the end points of its support (two jumps) and it has to be a piecewise constant function. Thus , for .

Some insight into degree of smoothness is given by the proposition ( Smoothness of compactly supported wavelets with vanishing moments ). More precise investigation is given in the source [Daubechies1992] .The grows of regularity with is rather slow. are continuously differentiable for . For large regularity grows like .

Remark

The following Mathematica script implements the procedure of the proposition ( Existence of smooth compactly supported wavelets ). The results agree with Python's pywt module.

n=3

CC[k_, n_] := Binomial[n, k]

Pnm1[n_, y_] :=

Expand[Sum[CC[k, 2*n - 1]*y^k*(1 - y)^(n - 1 - k), {k, 0, n - 1}]]

A[n_,z_]:=Sum[a[p]*Exp[-2*Pi*I*z*p],{p,0,n-1}]

Cond[n_,z_]:=Pnm1[n,(Sin[Pi*z])^2]-Conjugate[A[n,z]]*A[n,z]

m0[n_,z_]:=((1+Exp[-2*Pi*I*z])/2)^n*A[n,z]

x1 = Cond[n, z]

x2 = TrigToExp[ComplexExpand[x1]]

d = Exponent[x2, Exp[I*Pi*z]]

x3 = Collect[x2*Exp[d*I*Pi*z], Exp[I*Pi*z]]

L=CoefficientList[x3, Exp[I*Pi*z]]

eqs=Map[Function[x, x == 0], L]

eqs2 = Map[Function[x, Im[a[x]] == 0], Range[0, n-1]]

eqs3={ A[n,0]==1 }

vars = Map[Function[x, a[x]], Range[0, n-1]]

sol=NSolve[Join[eqs, eqs2,eqs3], vars]

x4 = m0[n, z] /. sol[[1]]

x5=Collect[x4, Exp[I*Pi*z]]

LL=CoefficientList[x5, Exp[-2*I*Pi*z]]

h=Map[Function[x,Sqrt[2]*x],LL]

 Notation. Index. Contents.