On the minimal norms of polynomial projections
For every \(n\in {\mathbb{N}}\) let \(\Pi_ n\) be the subspace of C[-1,1] which consists of all the algebraic polynomials of degree not greater than n, and denote by \({\mathcal P}_ n\) the set of projections from C[-1,1] onto \(\Pi_ n\). Also, let \(L_ n^{\min}\) denote any minimal projection in \({\mathcal P}_ n\), i.e. a projection whose norm is the smallest possible. The author remarks that next to finding minimal projections onto \(\Pi_ n\), which seems to be a very difficult problem, one could try at least to determine projections whose norms differ 'only a little' from \(\| L_ n^{\min}\|\). To this end one needs to look for lower bounds which will enable to prove the property (*) \(\lim_{n\to \infty}\| L_ n\| \cdot \| L_ n^{\min}\|^{-1}=0\) for some sequences \((L_ n)_{n\in {\mathbb{N}}}\) of projections. With this in mind the author states and proves the following theorem which is the main result in the paper: For every \(n\geq 2\) the norms of projections \(L_ n\in {\mathcal P}_ n\) are bounded as follows: \(\| L_ n\| \geq (4/\Pi^ 2)(\ln n-\ln \ln n).\) It is also proved, as a corollary of the above result, that the Chebyshev partial sum operators are asymptotically minimal, i.e. they satisfy property (*).
scientific article; zbMATH DE number 4214655