I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier transforms. With that limitation in mind, I am interested in the behavior of the following transform:

Given an $L_2$-integrable function $f:R\rightarrow R$, define

\[\overline{f}(a,b)=\int_{-\infty}^{\infty} \tanh(ax+b) f(x) dx\]

My questions are:

(1) Does anyone recognize this transform?

(2) Is it invertible? What is the inverse?

(3) Is it invertible given only $\overline{f}(1,b)$ for all $b\in R$?

(4) In my application, the functions $f$ are probability mass functions. If I wanted to add two random variables, I would convolve their pmfs, which can be done efficiently with a Fourier transform. Is there an equivalent operation that this transform enables?