Conditioning of Fourier sums on a quadratic curve

The stability property of representations of the Fourier sums on a quadratic curve is measured through a condition number. We analyze the conditioning of the continuous Fourier sums and the discrete Fourier sums on the unit circle and the standard parabola. We give an upper bound for the conditioning in terms of norms of two families of orthogonal polynomials in one variable. We establish explicit bounds for the conditioning of some special Fourier sums. In particular, we show that the conditioning of n th partial Fourier sums on the circle and the standard parabola corresponding to special weight functions grows at most like O ( n ).