On a theorem of Akhiezer
The authors extend Akhiezer's theorem about the relation between the Hilbert transforms of the function \(f(x) \in L_ 2 (R)\) and the function \(f(x)/(x-\gamma) \in L_ 2 (R)\), where \(\gamma\) is a constant on the functions from the weighted space \(L_ 1 (R,dt/(1+t^ 2))\). They assume also that \((f(x)-\alpha)/(t-\gamma)\) is a locally integrable function. Akhiezer's proof depends on calculations of Fourier transforms, using complex methods and therefore it was difficult to generalize this theorem for \(L_ p\)-space with \(p \neq 2\). The authors overcome these problems and provide the calculation of new examples of the Hilbert transforms of some functions. Considering rational weights for \(L_ 1\)-functions they obtain new interesting examples Akhiezer's type relations for the Hilbert transform.
scientific article; zbMATH DE number 561225