Integral equivalent to fitting a curve to a sum of functions

[admixtus]

Hello,

I am searching for some kind of transform if it is possible, similar to a Fourier transform, but for an arbitrary function.

Sort of an inverse convolution but with a kernel that varies in each point.

Or, like I say in the title of this topic a sort of continuous equivalent of fitting a curve to a sum of functions.

For example if I want to use Gaussians, I want to reproduce a function $$F(x)$$

As:

$$F(x) = \int \frac{f(y)}{\sqrt{4\pi t(y)}}e^{-\frac{(x-y)^2}{4 t(y)}} dy$$

Notice how t is a function of y.
This is easy for a finite sum of Gaussians with linear regression, but I'm searching for a continuous equivalent.

The closest thing that I found for Gausses is a Weierstrass transform. But the 'standard deviation' of the gausses doesn't vary in each point.

There are a ton of subjects that come close (linear regression, inverse convolution, Weierstrass transform,..) but they either are discrete or lack the variability of the convoluting kernel.

Does someone know a mathematical technique that can do this? Or know in what direction I have to look? Thanks!

 

[haruspex]

I'm not quite clear on what is given. Obviously F is given, and you want to find f, but how about t? Is t(y) a given function?

 

[admixtus]

I'm not quite clear on what is given. Obviously F is given, and you want to find f, but how about t? Is t(y) a given function?

Yes, t(y) and f(y) are functions that I want fo find, yes. Maybe I should have written it explicitly like that instead of implying it by saying the the kernel was variable.

 

[haruspex]

Yes, t(y) and f(y) are functions that I want fo find, yes. Maybe I should have written it explicitly like that instead of implying it by saying the the kernel was variable.

From my reading of the subject (totally new to me until I saw your post) the Weierstrass transform is exactly that, a transform, so is, generally speaking, invertible. This means there is not enough information to find t. Your mission would make more sense if t(y) were given. Am I missing something?

Not sure if this is what you are after, but look at the discussion of heteroscedastic Gaussian Processes at https://www.cs.cmu.edu/~andrewgw/andrewgwthesis.pdf