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After I have worked a little bit with different integral transform, especially with the Laplace transform, I was confronted with the fact that there are some functions which remain of the same type even after the transform.

For example the Gaussian Function in case of the Fourier transform or the reciprocal square root for the Laplace transform.

After a little bit of researching I figured out that these kind of functions are called fix points of the integral transforms and they satisfy the equation

$$(\mathcal{T}\varphi)(x)~=~\varphi(y) $$

where $\mathcal{T}$ denotes the transform and $\varphi$ the function which is a fix point of the given transform.

My question is about the identification of functions that might be a fix point to a given integral transform. Is there maybe a way to just look at the integral kernel and then to say which functions satisfy the above equation?

That there are many in some cases I saw when I was looking over a transform table (https://authors.library.caltech.edu/43489/1/Volume%201.pdf) I have found and some of these functions were not that obvious.

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  • $\begingroup$ Since you complain elsewhere that nobody's answered this: Is there maybe a way to just look at the integral kernel and then to say which functions satisfy the above equation? No. $\endgroup$ – David C. Ullrich Aug 11 '18 at 17:54
  • $\begingroup$ Thank you for your response here aswell. Again I do not know why you are using such a word as complain because I guess I wrote my other question in a proper way to avoid a provocation. Furthermore you could explain me why we cannot derive a fixed point from the kernel. Since every single fixed point is connected to the given kernel in some way I do not understand exactly why we cannot go the way backwards to derive a fixed point from the kernel. $\endgroup$ – mrtaurho Aug 12 '18 at 2:50
  • $\begingroup$ I didn't mean anything pejorative by the word "complain", sorry. When II say there's no way to derive a fixed point just by looking at $K$ I didn't mean I could formally prove there's no way, I meant just that it seems clear there's no way to do it. Can I explain why there's no way to do it? No. Can you explain how we would go backwards and find a fixed point? $\endgroup$ – David C. Ullrich Aug 12 '18 at 14:46
  • $\begingroup$ No, I cannot. Thats the main reason why I asked.^^' It seems so natural to me that one could at least guess what type of function has the potential to be a fixed point since these fixed points appear to be closely connected to the kernel. I have no idea how to approach further to this topic. I am just fascinated by the idea to transfer this well known concept of fixed point to the context of integral transform. Therefore I asked myself whether there is a pattern concerning these functions or not. I do not want to accept that it is just not possible - but maybe I have to :/ $\endgroup$ – mrtaurho Aug 12 '18 at 14:53
  • $\begingroup$ Well whether it seems natural to you or not, I have no idea how one would start, and evidently neither does anyone else who read this. $\endgroup$ – David C. Ullrich Aug 12 '18 at 15:08

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