Approximation properties of discrete Fourier sums

I. I. Sharapudinov

Diskretnaya Matematika, Tome 2 (1990) no. 2, pp. 33-44.

Résumé

Let $\omega^T_N=\{u+2j\pi /N\}^{N-1}_{j=0}$, $\omega_N=\{ 0,1,\cdots ,N-1\},$ where $u$ is an arbitrary real number and $2\leq N$ is a natural number. It is well known that the trigonometric functions $1$, $\cos x$, $\sin x,\dots,\cos nx$, $\sin nx$ ($2n\leqslant N$) form an orthogonal system on $\omega ^T_N$, and the Hahn polynomials $Q_0(x),\dots ,Q_{n-1}(x)$ form an orthogonal system on $\omega _N$ with weight $\rho (x)=\Gamma (x+\alpha +1)\Gamma (N-x+\beta )/(\Gamma (x+1)\Gamma (N-x))$, $\alpha,\beta>-1$. We study a problem on the approximation of discrete functions by Fourier sums with respect to these systems. We establish discrete analogues of the well-known result of K. I. Oskolkov on an estimate for the deviation of a Fourier partial sum of a continuous $2\pi$-periodic function.