Zur Minimalität trigonometrischer Polynomoperatoren

Abstract

Cheney and others [2] have shown that, with respect to the norm of uniform convergence, the Fourier operator F_n:C_2π→p_n is the only minimal projection. For a more detailed study of operators A:C_2π→P_n we investigate the evaluation functionals$\hat xAf: = (Af)(x)$ respectively the mean of their norms$\frac{1}{{2\pi }}\int\limits_0^{2\pi } {\left| {\hat xA} \right|dx}$. We give a complete characterization of polynomial operators which minimize this quantity. As an application we can simplify the proof in [2]. Moreover, we show that the trigonometric interpolation operators, having the above minimal-property,are exactly those with equidistant nodes.