Example 1 

By [Zhuang2010:complextTight], for any dilation d, sum rule order m, and vanishing moment n being odd,
there exits a symmetric, complex-valued, d-lowpass filter satisfies the sum rules of order m from which a tightframelet system
having vanishing moments n can be obtained.
 

> restart; 1
 

> read
 

> `assign`(d, 3); 1; `assign`(m, 5); 1; `assign`(n, 3); 1
`assign`(d, 3); 1; `assign`(m, 5); 1; `assign`(n, 3); 1
`assign`(d, 3); 1; `assign`(m, 5); 1; `assign`(n, 3); 1
 

 

 

3
5
3 (8.3.1)
 

> `assign`(py, D1GetPy(d, m, n)); 1
 

`+`(1, `*`(`/`(40, 3), `*`(y)), `*`(`/`(880, 9), `*`(`^`(y, 2)))) (8.3.2)
 

> `assign`(poly, D1ComplexTight(d, m, n)); 1
`assign`(poly, D1ComplexTight(d, m, n)); 1
 

`*`(`^`(`+`(`/`(1, 3), `*`(`/`(1, 3), `*`(z)), `*`(`/`(1, 3), `*`(`^`(z, 2)))), 5), `*`(`+`(`*`(`+`(`-`(`/`(5, 3)), `*`(`*`(`/`(1, 3), `*`(I)), `*`(`^`(30, `/`(1, 2))))), `*`(z)), `/`(13, 3), `-`(`*`(... (8.3.3)
 

> SYM(poly); 1
 

`*`(`^`(z, 10)) (8.3.4)
 

> `assign`(poly2, `/`(`*`(poly), `*`(`^`(z, 5)))); 1; SYM(poly2); 1
 

 

`/`(`*`(`^`(`+`(`/`(1, 3), `*`(`/`(1, 3), `*`(z)), `*`(`/`(1, 3), `*`(`^`(z, 2)))), 5), `*`(`+`(`*`(`+`(`-`(`/`(5, 3)), `*`(`*`(`/`(1, 3), `*`(I)), `*`(`^`(30, `/`(1, 2))))), `*`(z)), `/`(13, 3), `-`(...
1 (8.3.5)
 

> `assign`(n, 7); 1; `assign`(digits, 5); 1
`assign`(n, 7); 1; `assign`(digits, 5); 1
 

 

7
5 (8.3.6)
 

> `assign`(poly, D1ComplexTight(d, m, n, digits)); 1
 

`*`(`^`(`+`(`/`(1, 3), `*`(`/`(1, 3), `*`(z)), `*`(`/`(1, 3), `*`(`^`(z, 2)))), 5), `*`(`+`(`+`(`-`(62.354), `-`(`*`(13.877, `*`(I)))), `*`(`+`(2.5165, `-`(`*`(.72952, `*`(I)))), `*`(`^`(z, 3))), `*`(...
`*`(`^`(`+`(`/`(1, 3), `*`(`/`(1, 3), `*`(z)), `*`(`/`(1, 3), `*`(`^`(z, 2)))), 5), `*`(`+`(`+`(`-`(62.354), `-`(`*`(13.877, `*`(I)))), `*`(`+`(2.5165, `-`(`*`(.72952, `*`(I)))), `*`(`^`(z, 3))), `*`(...
`*`(`^`(`+`(`/`(1, 3), `*`(`/`(1, 3), `*`(z)), `*`(`/`(1, 3), `*`(`^`(z, 2)))), 5), `*`(`+`(`+`(`-`(62.354), `-`(`*`(13.877, `*`(I)))), `*`(`+`(2.5165, `-`(`*`(.72952, `*`(I)))), `*`(`^`(z, 3))), `*`(...
(8.3.7)
 

> SYM(poly); 1
 

`*`(`^`(z, 10)) (8.3.8)
 

> `assign`(poly2, `/`(`*`(poly), `*`(`^`(z, 5)))); 1; SYM(poly2); 1
 

 

`/`(`*`(`^`(`+`(`/`(1, 3), `*`(`/`(1, 3), `*`(z)), `*`(`/`(1, 3), `*`(`^`(z, 2)))), 5), `*`(`+`(`+`(`-`(62.354), `-`(`*`(13.877, `*`(I)))), `*`(`+`(2.5165, `-`(`*`(.72952, `*`(I)))), `*`(`^`(z, 3))), ...
`/`(`*`(`^`(`+`(`/`(1, 3), `*`(`/`(1, 3), `*`(z)), `*`(`/`(1, 3), `*`(`^`(z, 2)))), 5), `*`(`+`(`+`(`-`(62.354), `-`(`*`(13.877, `*`(I)))), `*`(`+`(2.5165, `-`(`*`(.72952, `*`(I)))), `*`(`^`(z, 3))), ...
`/`(`*`(`^`(`+`(`/`(1, 3), `*`(`/`(1, 3), `*`(z)), `*`(`/`(1, 3), `*`(`^`(z, 2)))), 5), `*`(`+`(`+`(`-`(62.354), `-`(`*`(13.877, `*`(I)))), `*`(`+`(2.5165, `-`(`*`(.72952, `*`(I)))), `*`(`^`(z, 3))), ...
1 (8.3.9)
 

>