Editing Hilbert transform (section)

== History ==
The Hilbert transform arose in Hilbert's 1905 work on a problem Riemann posed concerning analytic functions,{{sfn|Kress|1989}}{{sfn|Bitsadze|2001}} which has come to be known as the [[Riemann–Hilbert problem]]. Hilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle.{{sfn|Khvedelidze|2001}}{{sfn|Hilbert|1953}} Some of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave in [[Göttingen]]. The results were later published by Hermann Weyl in his dissertation.{{sfn|Hardy|Littlewood|Pólya|1952|loc=§9.1}} Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case.{{sfn|Hardy|Littlewood|Pólya|1952|loc=§9.2}} These results were restricted to the spaces [[Lp space|{{math|''L''<sup>2</sup>}} and {{math|ℓ<sup>2</sup>}}]]. In 1928, [[Marcel Riesz]] proved that the Hilbert transform can be defined for ''u'' in <math>L^p(\mathbb{R})</math> ([[Lp space|L<sup>p</sup> space]]) for {{math|1 < ''p'' < ∞}}, that the Hilbert transform is a [[bounded operator]] on <math>L^p(\mathbb{R})</math> for {{math|1 < ''p'' < ∞}}, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform.{{sfn|Riesz|1928}} The Hilbert transform was a motivating example for [[Antoni Zygmund]] and [[Alberto Calderón]] during their study of [[singular integral]]s.{{sfn|Calderón|Zygmund|1952}} Their investigations have played a fundamental role in modern harmonic analysis. Various generalizations of the Hilbert transform, such as the bilinear and trilinear Hilbert transforms are still active areas of research today.