Y transform

Remarks

Example: $Y_{1/2}(z) = -\sqrt{2/(\pi z)}\, \cos z$, because $Y_{1/2} = -J_{-1/2}$ and $J_{-1/2}(z) = \sqrt{2/(\pi z)}\, \cos z$. So $\sqrt{xy}\, Y_{1/2}(xy) = -\sqrt{2/\pi}\, \cos(xy)$, and $\mathcal{Y}_{1/2} f(y) = -\sqrt{2/\pi} \int_0^\infty f(x) \cos(xy)\, dx$, the negative of the Fourier cosine transform.

Non-example: for $\nu > 0$, $Y_\nu(z)$ behaves like $-\frac{\Gamma(\nu)}{\pi} (2/z)^{\nu}$ as $z \to 0^+$. For $\nu = 2$ and $f$ equal to $1$ on $(0, 1]$ and $0$ elsewhere, the integrand behaves like a constant times $x^{-3/2}$ near $0$, so the integral diverges for every $y > 0$.

The Hankel transform is defined in the same way with $J_\nu$ in place of $Y_\nu$.