Wavelets[1]

 

Periodic functions in 1-dimension

Assume for a complex function of a dimensionless real variable , the boundary condition

 

(1.1)

 

This is a transformation from the real line to the complex plane.

 

(1.2)

The Fourier series is

 

(1.3)

 

(1.4)

 

Orthogonality of the basis set of functions

 

(1.5)

 

Mother pre-wavelet of Fourier series is obviously

 

(1.6)

 

The basis is generated from it by integer dilation of the phase x.

 

Parseval identity of signal conservation of energy under basis transformations

 

(1.7)

 

 

 

The squeezed-shifted wavelet: Zoom in and zoom out!

We now allow non-periodic functions over the whole real line without the periodic boundary condition (1.1). Replace the sinusoidal wave mother pre-wavelet  by a localized mother wavelet packet . We must shift it to cover whole real line. One obvious way is to use integer shifts

 

(1.8)

 

We also need to dilate the mother wavelet. Communication engineers like to partition the frequencies into bands of consecutive “octaves” using integer powers of 2. The convention is

 

(1.9)

 

This is a combined operation of squeezing by a factor of  with a translation not of , but of rescaled . That is

 

(1.10)

 

Therefore,  squeezes the domain of support (where  ) into a smaller region as well as decreasing the shift. This is a zoom-out transformation like decreasing the magnification of a microscope showing less detail, less resolution of the image. On the contrary,  does the opposite zoom-in increasing the resolution of the image or increasing the magnification of the microscope. Note that ordinary Fourier analysis in physics does not have this capability that is particularly suitable for nonstationary statistical processes with fast changes especially for open non-equilibrium systems.

 

A wavelet series can be of the form

 

(1.11)

 

provided that we have a complete orthonormal basis[2] or “frame of reference”

 

(1.12)

 

In terms of quantum measurement theory, the squeezed-shifted wavelet base functions are formally “filters” or eigenfunctions of some “wavelet observable” if there is also a directly measurable eigenvalue by some constructable physical detector. This formal filter structure may or may not translate to something objectively real depending on more specific physical information.

 

Just like in Fourier analysis we can generalize a discrete integer wavelet series to a continuous integral wavelet transform where squeeze-shift integers j,k are replaced by real parameters a, b

 

(1.13)

 

We can, of course, still keep as a special case

 

(1.14)

 

consistent with (1.10). Note that the wavelet literature calls  the binary dilation, and  the dyadic position when the octave algorithm (1.14) is used.

 

“while the two components of Fourier analysis.. the Fourier series and the Fourier transform are basically unrelated; the two corresponding components of wavelet analysis, namely the wavelet series (1.11) and the integral wavelet transform (1.13) have an intimate relationship as described in (1.14)” p. 6 Chui

(1.15)