I have a set of n+1 functions $f_i:[0,1]\rightarrow R$, that I know are linearly dependent and I want to find the coefficients s.t. $\sum_i \alpha_i f_i=0$.
There are however three problems:
The fucntions are not differentiable (actually they are, but there's no way to find $f_i'$)
Evaluating the functions comes along with some numerical noise.
I do not have a closed analytical representation of the fucntions.
Essentially the functions are obtained by measuring real nature data.
Hence I can't use Wronskian and also can't just evaluate the fucntions at $n+1$ points and solve the linear equations, as the numerical errors arising from this will be to large. (Note that as I already have some numerical noise, the "real" coefficients $\alpha_i$ can not be found, but I want to find coefficients that do not have to much error in them).
One idea that I came up with is a sort of Fourier-series/orthogonal projection idea, but I think this can be improved.
Instead of looking for $\sum_i \alpha_i f_i =0$, we search teh equivalent equation $f_{n+1}=\sum_i\alpha_i f_i$.
We orthonormalize the functions $f_i$ for $i\leq n$ with gramschmidt (wrt. $<f,g> = \int_0^1f(x)\cdot(x)dx$) obtaining the functions $g_i$.
Then we write all the functions in this basis: $f_j = \sum_k \beta_{j,k} g_k$. From this we get linear equations (in $R^n$): $\sum_j \alpha_j \beta_{j,k} = 0 \ \forall k$. Which we can solve for $\alpha_j$. (As a bonus, solving this last linear equation is very easy, as the matrix $\beta = \{\beta_{j,k}\}_{j\leq n}$ has triangle form).