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.
 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.

Calculation of spline biorthogonal wavelets.

n this section we implement the procedure of the proposition ( Existence of biorthogonal compactly supported wavelets ) by taking (see the definition ( Spline functions )) for some . According to the proposition ( Properties of spline functions )-4, We seek to satisfy

For fixed , the are selected to insure that the highest powers of and would be the same on both sides of the equation . Therefore, we calculate the highest power of as follows thus or We calculate the highest power of : thus or

Remark

Note that the number of terms in the sequence is and the number of terms in the sequence is For purposes of the section ( Adapting GMRA to interval [0,1] ) we need Thus, we need or Therefore, to have vanishing moments and the same supports for , it suffices to introduce and with the following lengths:

Remark

(Unsuitability of spline wavelets) Note that in the expression the terms and partially cancel out. This means that instead of the and have representations for some . As a result, we cannot have both equal supports for and equal number of vanishing moments for and . Therefore, we are forced to seek alternative approach to factorization of the expression in the following sections.

Remark

(Unsuitability of spline wavelets 2)Obtained this way wavelets are not symmetrical with respect to regularity. The is set to be a spline. The function is wildly oscillating.

The procedure is implemented by the following Mathematica script. The results agree with the Python's pywt module and [Walnut] up to cubic spline wavelets (p<=2,n<=5).

n=2

p=2*n-2

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}]]

N1[p_,n_]:= p-2*n+2

N2[p_,n_]:= 2*n-1

m0[p_, x_] :=(1/2*(1+Exp[-2*Pi*I*x]))^(p+1)

M0[p_,n_,x_] := 1/Sqrt[2]*Sum[H[k]*Exp[-2*Pi*I*k*x], {k, N1[p,n],N2[p,n]}]

LHS[p_,n_,z_] := m0[p, z]*Conjugate[M0[p,n, z]]

RHS[n_, z_] := (Cos[Pi*z])^(2*n)*Pnm1[n, (Sin[Pi*z])^2]

Cond[p_,n_, z_] := Expand[TrigToExp[RHS[n, z]]] - TrigToExp[ComplexExpand[LHS[p,n,z]]]

x1 = Cond[p,n, z]

d=4*n

x2 = Collect[x1*Exp[d*I*Pi*z], Exp[I*Pi*z]]

L=CoefficientList[x2, Exp[I*Pi*z]]

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

eqs2 = Map[Function[x, Im[H[x]] == 0], Range[N1[p,n],N2[p,n]]]

eqs3 = { M0[p,n,0]==1 }

vars = Map[Function[x, H[x]], Range[N1[p,n],N2[p,n]]]

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

m0B[p_, x_] := 1/Sqrt[2]*Sum[h[k]*Exp[-2*Pi*I*k*x], {k, 0,p+1}]

CondB[p_,z_] := Expand[m0[p,z]-m0B[p,z]]

x1 = CondB[p,z]

x2 = Collect[x1, Exp[-I*Pi*z]]

L=CoefficientList[x2, Exp[-I*Pi*z]]

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

eqs2 = Map[Function[x, Im[h[x]] == 0], Range[0,p+1]]

vars = Map[Function[x, h[x]], Range[0,p+1]]

solB=NSolve[Join[eqs, eqs2], vars]

{solA,solB}

 Notation. Index. Contents.