Math Help - foorier series from haar function..

1. foorier series from haar function..

http://mathworld.wolfram.com/HaarFunction.html
$f(x)= {1, 0<=x<3/4}$
$f(x)= {1, 3/4<=x<=1}$
calculate the foorier function with haar group
here is the solution:
$L_0=\phi_{0,-1}(x)=1\\$
$L_1=\phi_{0,0}(x)=\begin{Bmatrix}1,0\leq x\leq 0.5
\\
-1,-0.5\leq x\leq 1
end{Bmatrix}\\$

i dont know why i have latex error here the code is
e_1=\phi_{0,0}(x)=\begin{Bmatrix}1,0\leq x\leq 0.5
\\
-1,-0.5\leq x\leq 1
end{Bmatrix}\\
$aL_0+bL_1\\$
$e_2=\phi{0,1}(x)\\$
$e_3=\phi{1,1}(x)\\$
$f=0.75\phi_{0,-1}(x)+1/4\phi_{0,0}(x)+1/2sqrtof2\phi_{1,1}(x)\\$
$=\int_{0}^{1}fe_0dx$

i dont know how they chose which haar function to take for the basis of of this serires
i could say that i take haar function with n=10
so my j=0..2^10
and i whould have much more e's

how they chose what haar sections to take as the orthonormal group
?

2. i have changed the function
there was a mistake