On polynomial ``interpolation'' in \(L_ 1\) (Q809299)

It is shown that every operator F from \(L_ 1[-\pi,\pi]\) into the space of trigonometric polynomials of degree \(m\geq n\) which satisfies n additional conditions \[ \int^{\pi}_{-\pi}g_ k(t)(Ff)(t)dt=\int^{\pi}_{-\pi}g_ k(t)f(t)dt,\quad k=1,2,...,n, \] where \(g_ k\in L_{\infty}[-\pi,\pi]\), supp \(g_ k\cap \sup p g_ j=\emptyset\) for \(j\neq k\), satisfies also \(\| F\|_{L_ 1\to L_ 1}\geq C \log (n/(m-n))\) with universal constant \(C>0\) independent of \(g_ 1,...,g_ n\).