General Procedure for Inverse of an Integral Transform - MathOverflow

General Procedure for Inverse of an Integral Transform

2012-04-09T19:59:21Z

Is there a general inversion formula or procedure for an integral of the form (where f is the function being transformed and g depends on the type of transform) $\int^{a}_{b} f(x) g(x,\xi) dx $ ? Inverses are defined in the conventional ways for functionals and integral transforms, respectively.

For instance, for the fourier transform. In the equation above, $a=∞,b=-∞$, $g(x,\xi)=e^{-2πix \xi }$. I know the inverse fourier transform is simple but I am concerned with a general procedure or process.

I am most interested in the cases where $a,b=±∞$ although a simple inverse for $\int^{a}_{b} f(x) dx $ is also something I am curious about (as far as this part of the question is concerned, if the inverse of an indefinite integral is the derivative, what is the inverse of a definite integral-- I am sorry if this is too elementary).

Feel free to use complex analysis or any other branch of math if it helps to answer the question. Also, you can repost on another site if it will help.

Answer by Philip van Reeuwijk for General Procedure for Inverse of an Integral Transform

2012-04-09T23:26:15Z

I think what Michael is saying in his comment is that the question is way too general: you are trying to deal with arbitrary linear operators on an infinite dimensional vector space, so invertibility (let alone a simple formula) is a subtle question that can't be answered in a simple fashion. You wonder about the inverse operation of definite integration over a fixed interval $I$; integration yields a real number (assuming the integral even exists!), so for $a\in\mathbb{R}$, "$\int_I^{-1}a = $ any function $f$ with $\int_If=a$", so to speak (sorry for the horrible notation). This is satisfied by loads of functions, so the inverse doesn't exist.

So if you have a specific operator in mind, you could try to see if it's invertible and if you can find a nice expression for the inverse. The general question doesn't make much sense.

Answer by Karmon Euloid for General Procedure for Inverse of an Integral Transform

2012-04-10T23:50:31Z

If we ignore the second part, I still think the first part has value. What is the procedure for inverting a general integral transform?

Answer by Tom Copeland for General Procedure for Inverse of an Integral Transform

2012-04-11T02:27:16Z

Just to direct you towards the literature and help you clear up some of your thinking on the subject take a look at Wiki's Inverse Problem. Then perhaps look at Methods of Applied Mathematics by Hildebrand and Principles and Techniques of Applied Mathematics by Friedman.

Answer by Dirk for General Procedure for Inverse of an Integral Transform

2012-04-11T07:56:22Z

Probably a "general procedure" would be as follows: Find appropriate Hilbert spaces $X$ and $Y$ such that the operator $Lf(\xi) \int_a^b f(x) g(x,\xi) dx$ maps boundedly from $X$ to $Y$. Often the space $X = L^2([a,b])$ and $Y= L^2([c,d])$ works (e.g. if $g\in L^2([a,b]\times [c,d])$). Then there always exists the Moore-Penrose pseudoinverse of $L$. However, if the range of $L$ is not a closed subspace of $y$, it is an unbounded operator defined on $\text{range}L\oplus \text{range}L^\bot$ which is a dense subspace of $Y$.

This works for $Lf = \int_a^b f(x)dx$ as a mapping from $L^2([a,b])$ to $\mathbb{R}$ and it is a nice exercise to work out the pseudo-inverse.