Can a Mellin / Laplace Transform-like method be done with functions beside $x^{s-1}, e^{-st}$?

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Was wondering if using other kernel functions beside these would result in illucidating other types of formulas than what the above two well-known transform methods typically handle.

I don't necessarily need an in depth explanation if you've tried it before. Just say "you'll run into trouble finding an inverse." etc.

Thanks.