Existence of smooth compactly supported wavelets. Daubechies polynomials.

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

2. Set where the $C_{n}^{k}$ are the binomial coefficients, .

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

4. Match MATH to find .

5. Set MATH

6. Use the proposition ( Cascade algorithm ) to find $\phi$ .

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

Remark

The step 4 is possible for the following reasons. We have MATH for some numbers and the sum is real-valued for $z\in\QTR{cal}{R}$ . Therefore the has to be real-valued by orthogonality of . We calculate MATH 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 MATH Thus MATH and the length of $\phi$ 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 auxiliary 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 MATH where the sequence is finite due to compactness of support and real because we seek to build an MRA with real valued $\phi$ and thus have the proposition ( Scaling equation ). The is sought to satisfy the following conditions: MATH MATH MATH MATH for some finite sequence of numbers .

We have MATH

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

Remark

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

Thus, the number of discontinuities of $\phi$ and its derivatives has to be finite (by compactness of support of $\phi$ in $\left( \#\right)$ ).

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. $\phi,\psi$ 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.