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The Laplace transform of a function $f(t)$, defined for all real numbers $t \geq 0$, is the function $F(s)$, defined by

$${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}.$$

Let $\varphi: {\mathbb R}_+ \to {\mathbb R}_+$ be an injective continuous increasing function (not necessarily linear). Is the function:

$${\displaystyle F_{\varphi}(s)=\int _{0}^{\infty }f(t)e^{- \varphi(st)}\,dt},$$

related to the Laplace Transform or another known integral transform?

After some calculations, I arrived to an integral of that kind, so I was wondering if some theoretical properties of that kind of functions are known already.

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A generalization along these lines, but with the single function $\phi(st)$ replaced by the product of two invertible functions $\Phi(s)E(t)$, has been studied in The generalized Laplace transform and fractional differential equations of distributed order. The Mellin transform is a notable example of this type, with $\Phi(s)E(t)=(1-s)\ln t$.

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  • $\begingroup$ Many thanks. I came across that paper, it looks interesting. I am starting to fear the generalisation I am considering has not been studied. $\endgroup$
    – Into
    Apr 27, 2020 at 20:48

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