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

Abstract.

For n -dimensional subspaces E _n , F _n of L ¹ (-1,1) with E _n spanned by Chebyshev polynomials of the second kind and F _n the set of Müntz 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 $\lambda$ (E _n , L ¹ (-1,1)) $\ge$ C log n and $\lambda$ (F _n , L ¹ (-1,1)) = O(1) , $n \to \infty$ . The spaces L ¹ _w(α,β) , where w _α,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 E _n , are not minimal.