Polynomial Approximation by Projections on the Unit Circle

Abstract

In the space $A(C)$ of functions continuous on the closed unit disc and analytic in interior points normed in the minimax sense, it is proved that the projection $T_n $ onto truncated Taylor series is a minimal projection onto polynomials. Moreover by computing a bound for $\| {T_n } \|$ it is shown that $T_n f$ is a practical near-minimax polynomial approximation to f in $A(C)$. The projection $F_n $ interpolating at the equally-spaced “Fourier points” on the unit circle, which is conjectured to be a minimal Lagrange interpolating projection, is shown to be a practical near-minimax polynomial approximation. The bounds on $\| {T_n } \|$ and $\| {F_n } \|$ have asymptotic order $({4 / {\pi ^2 }})\log n$ and $({2 / \pi })\log n$ respectively. Efficient algorithms for computing these two projections are based on the fast Fourier transform.