Conditioning of polynomial Fourier sums

In order to measure the stability properties of Fourier sums, we introduce a conditioning of a representation of the solution of the least squares problem. We relate the conditioning of the discrete polynomial Fourier sums with the corresponding continuous one. We study the asymptotic growth of the conditioning in terms of the degree n . For Fourier–Legendre sums, it is O ( n 3 / 2 ) . In the case of Fourier–Chebyshev sums, it is linear in the degree. For Fourier sums with Chebyshev polynomials of the second kind, it is O ( n 2 ) .