View source for Hardy transform

The integral transform

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where

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and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046390/h0463903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046390/h0463904.png" /> are the [[Bessel functions|Bessel functions]] of the first and second kinds, respectively. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046390/h0463905.png" /> the Hardy transform coincides with one of the forms of the Hankel transform, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046390/h0463906.png" /> with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046390/h0463907.png" />-transform. The Hardy transform was proposed by G.H. Hardy in [[#References|[1]]].

The inversion formula is

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where

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The Hardy transform is also defined for certain classes of generalized functions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   "Some formulae in the theory of Bessel functions"  ''Proc. London. Math. Soc. (2)'' , '''23'''  (1925)  pp. 61–63</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Y.A. Brychkov,   A.P. Prudnikov,   "Integral transforms of generalized functions" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR></table>