Hilbert transforms and Lagrange interpolation (Q914084)

This note completes the author's paper ``Mean convergence of Lagrange Interpolation III'' [Trans. Am. Math. Soc. 282, 669-698 (1984; Zbl 0577.41001)] where the formula \(\int_{{\mathbb{R}}}H(F)G=- \int_{{\mathbb{R}}}H(G)F\) (H is the Hilbert transform) is used. It is known that this formula is true for \(F\in L_ p\) and \(G\in L_ q\), \(1<p\), \(q<\infty\), \(1/p+1/q=1\). The author shows that for F and G having compact supports, \(F\in L \log^+L\) and \(G\in L_{\infty}\), the formula holds as well. The proof of this statement is due to Harold Widom.