Vol. 88, No. 1, 1980

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Fourier coefficients of the Rudin-Carleson extensions

Josip Globevnik

Vol. 88 (1980), No. 1, 69–79

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.

Mathematical Subject Classification 2000
Primary: 46J15
Secondary: 30H05
Received: 30 January 1979
Published: 1 May 1980
Josip Globevnik