### Approximation theory

### Daghestan Electronic Mathematical Reports, Issue 9 (2018)

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

### UDK: 519.688

### Pages: 1 - 6

### DOI: 10.31029/demr.9.1

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.

**Keywords: **
fast Fourier transform, discrete sine transform, inner product of Sobolev type, Sobolev orthogonal function.