Orthonormal functions as a combination of three complex functions

Question

I have some problem to find three orthonormal functions in the interval $-1\le x\le 1$ as a linear combination of these three functions: $$f_1(x)=1,f_2(x)=x\exp(i\pi x),f_3(x)=\exp(i\pi x)$$ Is it possible to use the Gram - Schmidt method? Thanks

Answer 1

Using Gram-Schmidt process, as correctly pointed out in the comments, you will notice that, in order to get three orthogonal functions in $L^2$ with the inner product $\langle f_i,f_j\rangle=\int_{-1}^1dxf_i^*(x)f_j(x)$, it is enough to rewrite $$ f_2(x)=xe^{i\pi x}-\frac{i}{\pi}. $$ Then you can properly normalize them.