If F is a closed subset of the
unit circle in C of Lebesgue measure 0 then by the Rudin-Carleson theorem every
continuous function f : F → C has a norm-preserving extension belonging to the disc
algebra. If we prescribe some Fourier coefficients of the extension g then in general
the norm of g will exceed the norm of f. In the paper we relate this problem to an
extremal problem and give optimal estimations of the norms of extensions with
prescribed Fourier coefficients. In particular, we give a precise description of those
functions f which have norm-preserving extensions g with finitely many prescribed
Fourier coefficients.
