This fact about convolutions of sincs is, I would say, widely known. Certainly it is at least to harmonic analysts and anyone doing signal processing. This is also how you define B-splines.

You are right, that is something I missed. If one modifies the definition to look at something like an iterated version of $\mathcal{H}$ then this I think is well defined, and possibly sufficient to consider. I will try to formulate it correctly and edit the question later.

Sorry I wrote that kind of off the cuff, I edited the formula to relay what I intended. This does look close to the Riesz transform, but slightly different.

Thanks for the reference, it has been a great starting point so far, and gave me some good references!