Near-Best Approximation by Averaging Polynomial Interpolants
Judith A. Palagallo, Thomas E. Price · IMA Journal of Numerical Analysis · 1987
Projections Mn−1,m are defined, which map the family A(C), of functions that are analytic in the unit disk and continuous on the boundary, to the space of polynomials of degree n−1 or less. A typical image polynomial is constructed by averaging m polynomials, each of which is induced by interpolating f ε A(C) at n equally spaced points on |z| = 1. An exact expression is obtained for the operator norm of Mn−1,m. The asymptotic behaviour of ‖Mn−1,m‖∞ is characterized as m→∞and n→∞. The results obtained generalize the classical results of Gronwall concerning interpolation at the roots of unity.