A lower bound for projection operators on \(L^ 1(-1,1)\) (Q1081765)

The following theorem is proved. For each \(n\in {\mathbb{P}}\) let \(P_ n\) be a bounded linear projection of \(L^ 1(-1,1)\) onto \({\mathcal P}_ n\), the set of algebraic polynomials of degree \(\leq n\). Then the operator norm of \(P_ n\) satisfies \[ \| P_ n\|_{[L^ 1]}\geq (2/\pi^ 2)\log n+O(1),\quad n\to \infty. \] In contrast to the classical Kharsiladze- Lozinski theorems, which deal with linear projections of the spaces C[- 1,1] and \(L^ 1_{\rho}(-1,1)\) with weight function \(\rho (x)=(1-x^ 2)^{-}\) or with the trigonometric analogues, this theorem is based on a new type of a so-called Berman-Marcinkiewicz identity which relates \(P_ n\) with the Chebyshev partial sum of the second kind. The novel idea is first to transform the problem in an appropriate way and second to employ a product formula of an unfamiliar sort, involving two different sets of orthogonal functions which may not be polynomials. As corollaries, analogues of the classical theorems of Nikolaev, Berman, and Faber are pointed out and an estimate from above and below of the relative projection constant \(\lambda\) (\({\mathcal P}_ n,L^ 1(-1,1))\) is given. We also prove theorems of the above kind for \(L^ 1\) spaces with Jacobi- weights \(w_{\alpha,\beta}(x)\), \((\alpha,\beta)=(0,-1/2)\) or \(=(- 1/2,0)\), and a further one on Fourier-Bessel projections.