POLYNOMIAL APPROXIMATION BY PROJECTIONS ON UNIT CIRCLE

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 the associated norm of T//n it is shown that T//nf 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. Efficient algorithms for computing these two projections are based on the fast Fourier transform.