This MAPLE 6 worksheet displays Haar's first two sons and first two
daughters simultaneously. It also uses the spreadsheet
package to display the inner product of these siblings.
Author: Dr. E. Rowe
Department of Mathematics
North Carolina A & T State Univ.
Greensboro, NC 27411
rowee@ncat.edu
Date: 13 Nov. 2000
>
> restart;
>
>
>
> phi := t -> piecewise(0 <= t and t < 1, 1);
>
> for n from 1 to 2 do
> for k from 0 to (2^n - 1) do
> phi||n||k := phi(2^n * t - k):
> phi||n||k := unapply(phi||n||k, t);
> end:
> end:
>
>
> with(plots):
Warning, the name changecoords has been redefined
>
>
> psi||1||0 := t -> phi(4*t) - phi(4*t - 1);
> psi||1||1 := t -> phi(4*t - 2) - phi(4*t - 3);
>
>
display(array(1..2,1..2,[[plot(phi||1||0(t),t=0..1, title=`Haar's First Son`), plot(phi||1||1(t), t=0..1, title=`Haar's Second Son`)],
[plot(psi||1||0(t),t=0..1, title=`Haar's First Daughter`), plot(psi||1||1(t), t=0..1, title=`Haar's Second Daughter`)]]));
![[Maple Plot]](images/rowe14.gif)
![phi[10]](images/rowe15.gif){kind=small}
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![phi[11]](images/rowe16.gif){kind=small}
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![psi[10]](images/rowe17.gif){kind=small}
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![psi[11]](images/rowe18.gif){kind=small}
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![phi[10]](images/rowe19.gif){kind=small}
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![phi[11]](images/rowe114.gif){kind=small}
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![psi[11]](images/rowe119.gif){kind=small}
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![psi[11]](images/rowe124.gif){kind=small}
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>
> with(Spread):
> mySpread := CreateSpreadsheet(Sons_and_Daughters);
>
>
>
>
> innerProduct := proc(f, g,a,b)
> # f - function
> # g - function
> # a - beginning of interval
> # b - end of interval
> int(f(x)*g(x),x=a..b);
> #evalf(%);
> end;
>
>
> SetCellFormula(mySpread,2,3,phi[10]);
> SetCellFormula(mySpread,2,4,phi[11]);
> SetCellFormula(mySpread,2,5,psi[10]);
> SetCellFormula(mySpread,2,6,psi[11]);
>
> SetCellFormula(mySpread,3,2,phi[10]);
> SetCellFormula(mySpread,4,2,phi[11]);
> SetCellFormula(mySpread,5,2,psi[11]);
> SetCellFormula(mySpread,6,2,psi[11]);
>
>
>
>
>
>
> SetCellFormula(mySpread,3,3,innerProduct(phi||1||0,phi||1||0,0,1)):
> SetCellFormula(mySpread,3,4,innerProduct(phi||1||0,phi||1||1,0,1)):
> SetCellFormula(mySpread,3,5,innerProduct(phi||1||0,psi||1||0,0,1)):
> SetCellFormula(mySpread,3,6,innerProduct(phi||1||0,psi||1||1,0,1)):
> SetCellFormula(mySpread,4,3,innerProduct(phi||1||1,phi||1||0,0,1)):
> SetCellFormula(mySpread,4,4,innerProduct(phi||1||1,phi||1||1,0,1)):
> SetCellFormula(mySpread,4,5,innerProduct(phi||1||1,psi||1||0,0,1)):
> SetCellFormula(mySpread,4,6,innerProduct(phi||1||1,psi||1||1,0,1)):
> SetCellFormula(mySpread,5,3,innerProduct(psi||1||0,phi||1||0,0,1)):
> SetCellFormula(mySpread,5,4,innerProduct(psi||1||0,phi||1||1,0,1)):
> SetCellFormula(mySpread,5,5,innerProduct(psi||1||0,psi||1||0,0,1)):
> SetCellFormula(mySpread,5,6,innerProduct(psi||1||0,psi||1||1,0,1)):
> SetCellFormula(mySpread,6,3,innerProduct(psi||1||1,phi||1||0,0,1)):
> SetCellFormula(mySpread,6,4,innerProduct(psi||1||1,phi||1||1,0,1)):
> SetCellFormula(mySpread,6,5,innerProduct(psi||1||1,psi||1||0,0,1)):
> SetCellFormula(mySpread,6,6,innerProduct(psi||1||1,psi||1||1,0,1)):
> EvaluateSpreadsheet(mySpread):
>
>
>
>