Wavelet and scaling functions

`[`

returns `phi`

,`psi`

,`xval`

] = wavefun(`wname`

,`iter`

)`psi`

and `phi`

, approximations of
the wavelet and scaling functions, respectively, associated with the orthogonal
wavelet `wname`

, or the Meyer wavelet. The approximations are
evaluated on the grid points `xval`

. The positive integer
`iter`

specifies the number of iterations
computed.

`[`

returns approximations of the wavelet and scaling functions associated with the
biorthogonal wavelet `phi1,psi1`

,`phi2,psi2`

,`xval`

] = wavefun(`wname`

,`iter`

)`wname`

. The wavelet and scaling
function approximations `psi1`

and `phi1`

,
respectively, are for decomposition. The wavelet and scaling function
approximations `psi2`

and `phi2`

,
respectively, are for reconstruction.

For compactly supported wavelets defined by filters, in general no closed form analytic formula exists.

The algorithm used is the cascade algorithm. It uses the single-level inverse wavelet transform repeatedly.

Let us begin with the scaling function ϕ.

Since ϕ is also equal to ϕ_{0,0}, this function is characterized
by the following coefficients in the orthogonal framework:

<ϕ, ϕ

> = 1 only if_{0,n}*n*= 0 and equal to 0 otherwise<ϕ, ψ

> = 0 for positive_{−j,k}*j*, and all*k*.

This expansion can be viewed as a wavelet decomposition structure. Detail coefficients are all zeros and approximation coefficients are all zeros except one equal to 1.

Then we use the reconstruction algorithm to approximate the function ϕ over a dyadic grid, according to the following result:

For any dyadic rational of the form *x* =
*n*2^{−j} in which the function is
continuous and where *j* is sufficiently large, we have pointwise
convergence and

where *C* is a constant, and α is a positive constant depending on
the wavelet regularity.

Then using a good approximation of ϕ on dyadic rationals, we can use piecewise constant or piecewise linear interpolations η on dyadic intervals, for which uniform convergence occurs with similar exponential rate:

So using a *J*-step reconstruction scheme, we obtain an approximation
that converges exponentially towards ϕ when *J* goes to
infinity.

Approximations are computed over a grid of dyadic rationals covering the support of the function to be approximated.

Since a scaled version of the wavelet function ψ can also be expanded on the
(ϕ_{−1,n)})* _{n}*,
the same scheme can be used, after a single-level reconstruction starting with the
appropriate wavelet decomposition structure. Approximation coefficients are all zeros
and detail coefficients are all zeros except one equal to 1.

For biorthogonal wavelets, the same ideas can be applied on each of the two multiresolution schemes in duality.

This algorithm may diverge if the function to be approximated is not continuous on dyadic rationals.

[1] Daubechies, I. *Ten
Lectures on Wavelets*. CBMS-NSF Regional Conference Series in Applied
Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics,
1992.

[2] Strang, G., and T. Nguyen.
*Wavelets and Filter Banks*. Wellesley, MA: Wellesley-Cambridge
Press, 1996.