Bases of trigonometric polynomials consisting of shifts of Dirichlet kernels

The system of shifts of Dirichlet kernel on $\frac{2k\pi}{2n+1}$, $k=0,\pm1,\dots,\pm n$, and the system of such shifts of the conjugate Dirichlet kernel with $\frac12$ are orthogonal bases in the space of trigonometric polynomials of degree $n$. The system of shifts of kernels $\sum_{k=m}^n \cos kx$ and $\sum_{k=m}^n\sin kx$ on $\frac{2k\pi}{n-m+1}$, $k=0,1,\dots,n-m$, is an orthogonal basis in the space of trigonometric polynomials with the components from $m\geqslant1$ tо $n$. There is no orthogonal basis of shifts of any function in this space for $0<m<n$.