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On the extension of a theorem of Stein and Weiss and its application. (English) Zbl 1071.42004
Summary: 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 $[0, 2\pi]$. As an application for the Hilbert transform of a bounded function, the optimal correlation $$\sup_{\Vert f\Vert_{L_\infty}\le 1}\Vert \widetilde f\Vert_L=4\widetilde K_1=4.664974464\dots,$$ where the Favard’s constant $\widetilde K_1$: $$\widetilde K_1=\frac 4\pi\ \sum^\infty_{\nu=0}\frac{(-1)^\nu} {(2\nu+1)^2}=\int^\infty_0\arctan\frac{1}{\sinh(\pi y/2)}\,dy,$$ is also established.

42A50Conjugate functions, conjugate series, singular integrals, one variable
28A25Integration with respect to measures and other set functions
30E25Boundary value problems, complex analysis
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