0.5 The scaling function and scaling coefficients, wavelet and  (Page 5/13)

holds.

This condition, called the fundamental condition [link] , [link] , gives a slightly tighter result than Theorem  [link] . While the scaling function still may be a distribution not in $L^1$ or $L^2$ , it is better behaved than required by Theorem  [link] in being defined on the dense set of dyadic rationals. This theorem is equivalent to requiring $H\left(\pi \right)=0$ which from the product formula [link] gives a better behaved $\text{Φ}\left(\omega \right)$ . It also guarantees a unity eigenvalue for $\mathbf{M}$ and $\mathbf{T}$ but not that other eigenvalues do not exist with magnitudes larger than one.

The next several theorems use the transition matrix $\mathbf{T}$ defined in [link] which is a down-sampled autocorrelation matrix.

Theorem 11 If the transition matrix $\mathbf{T}$ has eigenvalues on or in the unit circle of the complex plane and if any on the unit circle are multiple, they have acomplete set of eigenvectors, then $\phi \left(t\right)\in L^2$ .

If $T$ has unity magnitude eigenvalues, the successive approximation algorithm (cascade algorithm) [link] converges weakly to $\phi \left(t\right)\in L^2$ [link] .

Theorem 12 If the transition matrix $T$ has a simple unity eigenvalue with all other eigenvalues having magnitude less than one, then $\phi \left(t\right)\in L^2$ .

Here the successive approximation algorithm (cascade algorithm) converges strongly to $\phi \left(t\right)\in L^2$ . This is developed in [link] .

If in addition to requiring [link] , we require the quadratic coefficient conditions [link] , a tighter result occurs which gives $\phi \left(t\right)\in L^2\left(\mathbf{R}\right)$ and a multiresolution tight frame system.

Theorem 13 (Lawton) If $h\left(n\right)$ has finite support or decays fast enough and if ${\sum }_{n}h\left(n\right)=\sqrt{2}$ and if ${\sum }_{n}h\left(n\right)h(n-2k)=\delta \left(k\right)$ , then $\phi \left(t\right)\in L^2\left(\mathbf{R}\right)$ exists, and generates a wavelet system that is a tight frame in $L^2$ .

This important result from Lawton [link] , [link] gives the sufficient conditions for $\phi \left(t\right)$ to exist and generate wavelet tight frames. The proof uses an iteration of the basic recursion equation [link] as a successive approximation similar to Picard's method for differential equations. Indeed, this method is used to calculate $\phi \left(t\right)$ in [link] . It is interesting to note that the scaling function may be very rough, even “fractal" in nature. This maybe desirable if the signal being analyzed is also rough.

Although this theorem guarantees that $\phi \left(t\right)$ generates a tight frame, in most practicalsituations, the resulting system is an orthonormal basis [link] . The conditions in the following theorems are generally satisfied.

Theorem 14 (Lawton) If h(n) has compact support, ${\sum }_{n}h\left(n\right)=\sqrt{2}$ , and ${\sum }_{n}h\left(n\right)h(n-2k)=\delta \left(k\right)$ , then $\phi (t-k)$ forms an orthogonal set if and only if the transition matrix $T$ has a simple unity eigenvalue.

This powerful result allows a simple evaluation of $h\left(n\right)$ to see if it can support a wavelet expansion system [link] , [link] , [link] . An equivalent result using the frequency response of the FIR digitalfilter formed from $h\left(n\right)$ was given by Cohen.

Theorem 15 (Cohen) If $H\left(\omega \right)$ is the DTFT of $h\left(n\right)$ with compact support and ${\sum }_{n}h\left(n\right)=\sqrt{2}$ with ${\sum }_{n}h\left(n\right)h(n-2k)=\delta \left(k\right)$ ,and if $H\left(\omega \right)\ne 0$ for $-\pi /3\le \omega \le \pi /3$ , then the $\phi (t-k)$ satisfying [link] generate an orthonormal basis in $L^2$ .

A slightly weaker version of this frequency domain sufficient condition is easier to prove [link] , [link] and to extend to the M-band case for the case of no zerosallowed in $-\pi /2\le \omega \le \pi /2$ [link] . There are other sufficient conditions that, together with those in Theorem [link] , will guarantee an orthonormal basis. Daubechies' vanishing moments will guaranteean orthogonal basis.