Bases of trigonometric polynomials consisting of shifts of Dirichlet kernels

Abstract

The system of shifts of Dirichlet kernel on \(\frac{{2k\pi }} {{2n + 1}} \), k = 0, ± 1, …, ± n, and the system of such shifts of conjugate Dirichlet kernels with \(\frac{1} {2} \) are orthogonal bases in the space of trigonometric polynomials of degree n. The system of shifts of the kernels \(\Sigma _{k = m}^n \) cos kx and \(\Sigma _{k = m}^n \) sin kx on \(\frac{{2k\pi }} {{n - m + 1}} \), k = 0, 1, …, n−m, is an orthogonal basis in the space of trigonometric polynomials with the components from m ⩾ 1 to n. There is no orthogonal basis of shifts of any function in this space for 0 < m < n.