Originally Posted by

**CaptainBlack** 1. Are you sure that you have not lost a factor of $\displaystyle \sqrt{2}$ somewhere?

2. For any complete ortho-normal basis $\displaystyle \{\phi_i; \ i=1,2, ..\}$ for an appropriate class of functions on $\displaystyle (a,b)$ (with respect to some weight $\displaystyle w$ if we want) we have:

$\displaystyle

f(x)=\sum_{k=0}^{\infty} a_n \phi_n(x)

$

where:

$\displaystyle

a_n=\int_a^b f(x) \phi_n(x) w(x) ~dx

$

Now the basis functions in you post are certainly orthogonal, and with $\displaystyle w(x)=2$ are orthonormal on $\displaystyle (0,1)$, so we need only show that they are a complete basis (which I don't have time to do at present, I may come back to this).

RonL