Bounds for Relative Projection Constants in L 1 (-1,1)

Abstract

For n -dimensional subspaces En, Fn of L1(-1,1) with En spanned by Chebyshev polynomials of the second kind and Fn the set of Muntz polynomials \(\sum_{j=1}^n a_j x^{m^j}\) with \(m \in {\bf N}\) , \(m \ge 8\) , it is shown that the relative projection constants satisfy \(\lambda\)(En, L1(-1,1))\(\ge\)C log n and \(\lambda\)(Fn, L1(-1,1)) = O(1) , \(n \to \infty\) . The spaces L1w(α,β) , where wα,β is the weight function of the Jacobi polynomials and \((\alpha,\beta) \in \{ (-\frac{1}{2},-\frac{1}{2}),(-\frac{1}{2},0),(0,-\frac{1}{2}) \}\) , are also studied. The Jacobi partial sum projections, which are used in connection with En , are not minimal.