Mehler–Fock transform

Remarks

Under suitable conditions on $f$, the inversion formula is $$f(x) = \int_0^\infty \tau \tanh(\pi \tau)\, P_{-1/2 + i\tau}(x)\, F(\tau)\, d\tau, \qquad x > 1.$$ The function $P_{-1/2 + i\tau}(x)$ is real for real $\tau$ and $x \ge 1$, and it decays like $x^{-1/2}$ up to a logarithmic factor as $x \to \infty$.

Example: if $f$ is integrable and vanishes outside $[1, 2]$, then the integral is over a compact interval of a continuous kernel, so $F(\tau)$ is defined for every $\tau \ge 0$.

Non-example: $f(x) = x$ has no Mehler–Fock transform, since $x P_{-1/2 + i\tau}(x)$ grows like $x^{1/2}$ up to a logarithmic factor and is not integrable on $[1, \infty)$.

The substitution $x = \cosh t$ turns the transform into an integral over $t \in [0, \infty)$, the form used on the hyperbolic plane.