On the extension of a theorem of Stein and Weiss and its application

Abstract

In this paper there is proved a generalization of a theorem of Stein and Weiss concerning metric properties of the conjugate characteristic functions of given sets on the interval [0, 2π]. As an application for the Hilbert transform of a bounded function, the optimal correlation

where the Favard's constant

:

is also established.

Keywords:

• Fourier series
• Conjugate characteristic functions
• Lebesgue's measure
• Optimal correlation
• Upper estimation
• 1991 Mathematics Subject Classification: Primary 42A50
• Secondary 28A25