On the stability of the representation of finite rank operators

Abstract

The stability of the representation of finite rank operators in terms of a basis is analyzed. A conditioning is introduced as a measure of the stability properties. This conditioning improves some other conditionings because it is closer to the Lebesgue function. Improved bounds for the conditioning of the Fourier sums with respect to an orthogonal basis are obtained, in particular, for Legendre, Chebyshev, and disk polynomials. The Lagrange and Newton formulae for the interpolating polynomial are also considered.