Algorithm for numerical realization of polynomials in functions orthogonal in the sense of Sobolev and generated by cosines

In this paper we developed an algorithm for numerical computation of polynomials by the functions $\xi_{1,0}(t)=1,\ \xi_{1,1}(t)=t,\ \xi_{1,n+1}(t)=\frac{\sqrt{2}}{\pi n}\sin(\pi nt),\ (n=1,2,\ldots)$ on the grid $\{t_j=\frac{j}{N}\}_{j=0}^{N-1}$. These functions are orthogonal on Sobolev with respect to the inner product $\langle f, g\rangle=f(0)g(0)+\int_0^1f'(t)g'(t)dt$ and generated by functions $\xi_0(x)=1,\ \{\xi_n(t)=\sqrt{2}\cos(\pi nt)\}_{n=1}^\infty$. The algorithm is based on the fast Fourier transform.