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This is kind of a second part of this question: How do Integral Transforms work. There, I replied a comment asking about how can I find kernel for the inverse integral transform and then I was encouraged to create a new question.

In fact, it is about how Kernels work. For example, let's suppose I have created an Integral Transform whose Kernel is:

$$K(s,t) = \sqrt{s+t}$$ or $$K(s,t) = \ln(s+t)$$

How can I find/calculate its inverse? Or more generally: given a Kernel, how should I proceed to get its kernel for the inverse integral transform?

Also, what are the possibilities of creating a Kernel, i.e. what should $K$ satisfy? Additionally, what are the advantages of Laplace's Kernel ($e^{-st}$)?

Which books, papers or anything would you recommend me to read? I only know Calculus (Single-Variable, Multi-Variable and Vectorial), a bit of ODEs and some Linear Algebra. Thanks

Mr. N
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  • No inverse exists for $K=\sqrt{st}$... – Lee Feb 20 '20 at 06:07
  • Frankly speaking, there are two fact came to my mind. First one is "kernel for the inverse integral transform" (it may be Laplace or Fourier or something else) and the other is "inverse kernel". I have no idea about whether the phrase "inverse kernel" exists or not (for the such cases) and also if it exists, then how to deal with it. But if it is "kernel for the inverse integral transform" then there are ways. – nmasanta Feb 20 '20 at 10:40
  • @nmasanta Sorry, I meant "kernel for the inverse integral transform". Could you explain me these mentioned ways? – Mr. N Feb 20 '20 at 13:37
  • @Szeto why it does not exist? – Mr. N Feb 20 '20 at 13:37
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    @Mr.N You can just factor out $\sqrt s$...so the transform is essentially $\sqrt s\int^b_a f(t)\sqrt t dt$. Clearly you can construct examples of $f$ to show that this transform is not injective. – Lee Feb 20 '20 at 14:05
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    @Szeto oh yeah sure. Then this is one requirement for creating a Kernel. Thanks – Mr. N Feb 20 '20 at 14:08

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