Hardy transform

The integral transform

$$F(x)=\int\limits_0^\infty C_\nu(xt)tf(t)dt,$$

where

$$C_\nu(z)=\cos p\pi J_\nu(z)+\sin p\pi Y_\nu(z),$$

and $J_\nu(z)$ and $Y_\nu(z)$ are the Bessel functions of the first and second kinds, respectively. For $p=0$ the Hardy transform coincides with one of the forms of the Hankel transform, and for $p=1/2$ with the $Y$-transform. The Hardy transform was proposed by G.H. Hardy in [1].

The inversion formula is

$$f(t)=\int\limits_0^\infty\Phi(tx)xF(x)dx,$$

where

$$\Phi(x)=\sum_{n=0}^\infty\frac{(-1)^n(x/2)^{\nu+2p+2n}}{\Gamma(p+n+1)\Gamma(\nu+p+n+1)}.$$

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

References

[1]	G.H. Hardy, "Some formulae in the theory of Bessel functions" Proc. London. Math. Soc. (2) , 23 (1925) pp. 61–63
[2]	Y.A. Brychkov, A.P. Prudnikov, "Integral transforms of generalized functions" , Gordon & Breach (1989) (Translated from Russian)