Integral approximation of the characteristic function of an interval by trigonometric polynomials

A. G. Babenko ; Yu. V. Kryakin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 14 (2008) no. 3, pp. 19-37

Résumé

We prove that the value $E_{n-1}(\chi_h)_L$ of the best integral approximation of the characteristic function $\chi_h$ of an interval $(-h,h)$ on the period $[-\pi,\pi)$ by trigonometric polynomials of degree at most $n-1$ is expressed in terms of zeros of the Bernstein function $\cos\{[nt-\arccos2q-(1+q^2)\cos t]/(1+q^2-2q\cos t)\}$, $t\in[0,\pi]$, $q\in(-1,1)$. Here, the parameters $q$, $h$, and $n$ are connected in a special way; in particular, $q=\sec h-\operatorname{tg} h$ при $h=\pi/n$.