Haar Wavelets
> #   the Haar scaling function  phi(x)
> phi := x -> piecewise( 0 < x and x <= 1, 1 );
> 
> plot(phi(x), x=-1..2 );
> 

              phi := x -> piecewise(0 < x and x <= 1, 1)


> # phi(t) = phi(2*t) + phi(2*t-1)
> plot( t -> phi(2*t) + phi(2*t-1), -1..2 );
> 
> 

> phi_ := (i,j) -> subs(I=i,J=j, x -> 2^(J/2) * phi( 2^J * x - I ) );
> 

                                             (1/2 J)      J
  phi_ := (i, j) -> subs(I = i, J = j, x -> 2        phi(2  x - I))

> plot( {phi_(i,1)(x) $ i=0..1}, x=-1..2 );
> 

> plot( x -> phi_(0,1)(x) + phi_(1,1)(x), -1..2 );

> plot( {phi_(i,2) $ i=0..3}, -1..2 );
> 
> 

> innerproduct := (f,g) -> int( f(x) * g(x), x=0..1 );
> 

                                          1
                                         /
                                        |
             innerproduct := (f, g) ->  |   f(x) g(x) dx
                                        |
                                       /
                                         0

> innerproduct( phi_(0,2), phi_(0,2) );
> innerproduct( phi_(0,2), phi_(1,2) );
> innerproduct( phi_(0,2), phi_(2,2) );
> innerproduct( phi_(0,2), phi_(3,2) );
> 

                                  1


                                  0


                                  0


                                  0

> f := 3*phi_(0,2) + 1*phi_(1,2) + 2*phi_(2,2) + 1*phi_(3,2);
> 
> innerproduct( f, phi_(0,2) );
> innerproduct( f, phi_(1,2) );
> innerproduct( f, phi_(2,2) );
> innerproduct( f, phi_(3,2) );
> 
> 

  f := 3 (x -> 2 phi(4 x)) + (x -> 2 phi(4 x - 1))

         + 2 (x -> 2 phi(4 x - 2)) + (x -> 2 phi(4 x - 3))


                                  3


                                  1


                                  2


                                  1

> # The Haar wavelets
> Haar_Psi := x -> piecewise( 0 < x and x <= 1/2, 1, 1/2 < x and x <= 1, -1 );


  Haar_Psi :=

        x -> piecewise(0 < x and x <= 1/2, 1, 1/2 < x and x <= 1, -1)

> plot(Haar_Psi(x), x=-1..2 );

> # Haar_Psi(t) =  phi(2*t) - phi(2*t-1)
> 
> plot( t -> phi(2*t) - phi(2*t-1), -1..2 ); 

> Psi_ := (i,j) -> subs(I=i,J=j, x -> 2^(J/2) * Haar_Psi( 2^J * x - I ) );

  Psi_ := (i, j) ->

                                 (1/2 J)           J
        subs(I = i, J = j, x -> 2        Haar_Psi(2  x - I))

> plot( {Psi_(i,2)(x) $ i=0..3}, x=-1..2 );

> #  Psi(t) = phi(2*t) - phi(2*t-1)
> 
> plot( t -> phi(2*t) - phi(2*t-1),  -1..2 );

> g := 5*Psi_(0,2) + 2*Psi_(1,2) - 3*Psi_(2,2) + 1*Psi_(3,2);

  g := 5 (x -> 2 Haar_Psi(4 x)) + 2 (x -> 2 Haar_Psi(4 x - 1))

         - 3 (x -> 2 Haar_Psi(4 x - 2)) + (x -> 2 Haar_Psi(4 x - 3))

> plot( g, 0..1 );

> innerproduct := (f,g) -> int( f(x) * g(x), x=0..1 );
> 

                                          1
                                         /
                                        |
             innerproduct := (f, g) ->  |   f(x) g(x) dx
                                        |
                                       /
                                         0

> innerproduct( g, Psi_(0,2) );

                                  5

> innerproduct( g, Psi_(1,2) );

                                  2

> innerproduct( g, Psi_(2,2) );

                                  -3

> innerproduct( g, Psi_(3,2) );

                                  1

> innerproduct( f, Psi_(0,1) );
> innerproduct( f, Psi_(1,1) );

                                  1/2
                                 2


                                    1/2
                               1/2 2

> innerproduct( f, Psi_(0,0) );

                                 1/2

>