Orthogonal and positive basis function with support $[0,1]$

Question

I have a problem in my research where I have to use a linear combination of the form

$$f(u)=\sum_{i=1}^{M}\omega_{i}f_i(u)$$

I have some constraints on the basis functions $f_i$:

• $f_i(u)\geq0$
• to have support on $[0,1]$
• to be orthogonal ($\langle f_i, f_j \rangle = 0$ for $i\neq j$ and $\langle f_i, f_j \rangle = c$ for $i=j$).

I use at the moment Bernstein basis, but it is not orthogonal. I used the method from Bellucci (2014) to make the Bernstein basis orthogonal, but I lose the positivity constraint...

Does anyone have an idea of such a basis? It doesn't have to be a polynomial basis.