I have been doing a fair amount of research into a complex analytic modified version of the Mellin transform. I have hit a few roadblocks, and am hoping there may already be literature on the subject. And it is not the transform itself that is my interest but rather specific functions under the transform.
I was inspired more or less by the Paley-Wiener theorems on the Fourier transform. Namely, instead of considering $f(x) : \mathbb{R} \to \mathbb{R}$ such that $\int_{-\infty}^{\infty} |f(x)|\,dx < \infty$ we consider holomorphic $f(z) $ in the strip $\Im(z) < \tau$ where $\int_{-\infty}^{\infty} |f(x+it)|\,dx < \infty$ for $ -\tau < t < \tau$. And then look closely at the Fourier transform acting on this function. This allows for simplicity of inverting the Fourier transform, obtaining asymptotics, proving Poisson's summation formula, and a few other results about the Fourier transform we get for free.
The following space $\mathcal{D_\theta}$ I'm interested in is similar to a weighted $L^1(0,\infty)$ but is slightly modified. If $f \in \mathcal{D}_{\theta}$ then $f(z)$ is holomorphic for $|arg(z)| < \theta$. And further $\int_0^\infty |f(xe^{it})|x^{-\sigma}\,dx < \infty$ for $-\theta < t < \theta$ and all $0 < \sigma < 1$. This has benefits when considering the Mellin transform, rather than simply considering $f$ such that $\int_0^\infty |f(x)|x^{-\sigma}\,dx < \infty$
I've been able to show a similar result to the Paley-Wiener theorem on asymptotics of the Mellin Transform of these types of functions. Let $\mathcal{M} f = F(z)$ be the Mellin transform of $f$ paired with its inverse $f = \mathcal{M}^{-1}F$. $F$ is holomorphic for $0 <\Re(z) < 1$ and for all $\theta>t > 0$ there exists a constant $M_t$ such that
$$|F(z)|< M_t e^{-t|\Im(z)|} \Leftrightarrow f \in \mathcal{D}_{\theta}$$
Recalling
$$F(z) = \int_0^\infty f(x)x^{z-1}\,dx$$
I've also uncovered a few convergence lemmas that are much simpler. Namely showing that $f_n \in L^1(0,\infty)$ converges to $f \in L^1(0,\infty)$ requires showing $||f_n - f||_{L^1} \to 0$ as $n\to\infty$, however showing $f_n \in \mathcal{D}_{\theta}$ converges to $f \in \mathcal{D}_{\theta}$ is simpler and needs not mention the $L^1$ norm at all.
My question is if anyone has any references to papers on the space $\mathcal{D}_\theta$, a complex modification of a weighted $L^1(0,\infty)$ space. I figure the space is similar to some kind of Hardy space however I've yet to see one like the above.
The reason I am interested in this space is rather difficult to describe but involves a lot of "unconventional analysis." Any questions, comments, suggestions are greatly welcomed. I am hoping I am not the only one who has considered this type of transform.
