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[link] gives the same information for the length-6, 4, and 2 Daubechies scaling coefficients, wavelet coefficients, scaling coefficient moments,and wavelet coefficient moments. Again notice how many discrete wavelet moments are zero.

[link] shows the continuous moments of the scaling function φ ( t ) and wavelet ψ ( t ) for the Daubechies systems with lengths six and four. The discrete moments are the momentsof the coefficients defined by [link] and [link] with the continuous moments defined by [link] and [link] calculated using [link] and [link] with the programs listed in Appendix C.

Daubechies Scaling Function and Wavelet Coefficients plus their Moments
Daubechies N = 6
n h ( n ) h 1 ( n ) μ ( k ) μ 1 ( k ) k
0 0.33267055295008 -0.03522629188571 1.414213 0 0
1 0.80689150931109 -0.08544127388203 1.155979 0 1
2 0.45987750211849 0.13501102001025 0.944899 0 2
3 -0.13501102001025 0.45987750211849 -0.224341 3.354101 3
4 -0.08544127388203 -0.80689150931109 -2.627495 40.679682 4
5 0.03522629188571 0.33267055295008 5.305591 329.323717 5
Daubechies N = 4
n h ( n ) h 1 ( n ) μ ( k ) μ 1 ( k ) k
0 0.48296291314453 0.12940952255126 1.414213 0 0
1 0.83651630373781 0.22414386804201 0.896575 0 1
2 0.22414386804201 -0.83651630373781 0.568406 1.224744 2
3 -0.12940952255126 0.48296291314453 -0.864390 6.572012 3
Daubechies N = 2
n h ( n ) h 1 ( n ) μ ( k ) μ 1 ( k ) k
0 0.70710678118655 0.70710678118655 1.414213 0 0
1 0.70710678118655 -0.70710678118655 0.707107 0.707107 1
Daubechies Scaling Function and Wavelet Continuous and Discrete Moments
N = 6
k μ ( k ) μ 1 ( k ) m ( k ) m 1 ( k )
0 1.4142135 0 1.0000000 0
1 1.1559780 0 0.8174012 0
2 0.9448992 0 0.6681447 0
3 -0.2243420 3.3541019 0.4454669 0.2964635
4 -2.6274948 40.6796819 0.1172263 2.2824642
5 5.3055914 329.3237168 -0.0466511 11.4461157
N = 4
k μ ( k ) μ 1 ( k ) m ( k ) m 1 ( k )
0 1.4142136 0 1.0000000 0
1 0.8965755 0 0.6343975 0
2 0.5684061 1.2247449 0.4019238 0.2165063
3 -0.8643899 6.5720121 0.1310915 0.7867785
4 -6.0593531 25.9598790 -0.3021933 2.0143421
5 -23.4373939 90.8156100 -1.0658728 4.4442798

These tables are very informative about the characteristics of wavelet systems in general as well as particularities of the Daubechies system.We see the μ ( 0 ) = 2 of [link] and [link] that is necessary for the existence of a scaling function solution to [link] and the μ 1 ( 0 ) = m 1 ( 0 ) = 0 of [link] and [link] that is necessary for the orthogonality of the basis functions. Ortho normality requires [link] which is seen in comparison of the h ( n ) and h 1 ( n ) , and it requires m ( 0 ) = 1 from [link] and [link] . After those conditions are satisfied, there are N / 2 - 1 degrees of freedom left which Daubechies uses to set wavelet moments m 1 ( k ) equal zero. For length-6 we have two zero wavelet moments and for length-4, one. For all longer Daubechies systems we have exactly N / 2 - 1 zero wavelet moments in addition to the one m 1 ( 0 ) = 0 for a total of N / 2 zero wavelet moments. Note m ( 2 ) = m ( 1 ) 2 as will be explained in [link] and there exist relationships among some of the values of the even-ordered scaling functionmoments, which will be explained in [link] through [link] .

As stated earlier, these systems have a maximum number of zero moments of the wavelets which results in a high degree ofsmoothness for the scaling and wavelet functions.  [link] and  [link] show the Daubechies scaling functions and wavelets for N = 4 , 6 , 8 , 10 , 12 , 16 , 20 , 40 . The coefficients were generated by the techniques described in Section: Parameterization of the Scaling Coefficients and Chapter: Regularity, Moments, and Wavelet System Design . The Matlab programs are listed in Appendix C and values of h ( n ) can be found in [link] or generated by the programs. Note the increasing smoothness as N is increased. For N = 2 , the scaling function is not continuous; for N = 4 , it is continuous but not differentiable; for N = 6 , it is barely differentiable once; for N = 14 , it is twice differentiable, and similarly for longer h ( n ) . One can obtain any degree of differentiability for sufficiently long h ( n ) .

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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