# How to find the inverse of an integral transform?

• I
• hyurnat4
In summary, the conversation discusses a problem of finding the distribution of a random variable ##T## subject to a condition involving its expected value. The problem can be solved by finding the inverse transform, but the function ##V## involved is not well-behaved. The speaker is seeking advice on how to find the inverse transform or alternative methods for solving the problem.

#### hyurnat4

I'm trying to find the distribution of a random variable ##T## supported on ##[t_1, t_2]## subject to ## \mathbb{E}[V(t', T)] = K, \forall t' \in [t_1, t_2]##. In integral form, this is : $$\int_{t_1}^{t_2} V(t', t).f(t) \, dt = K,\forall t' \in [t_1, t_2],$$ which is just an exotic integral transform.

So if I can find the inverse transform, I'm done. But the function ##V## is ...not nice. It's nowhere differentiable for ##t < t'##, there's a jump at ##t = t'## and it's constant from then on.

So do you guys know of any methods to find the inverse to an integral transform with a nasty kernel like ##V##? Or can you see another way to solve these equations? I'd even be happy with numerical techniques.

##V(t^\prime,t)## is similar to Brownian motion for ##t<t^\prime##?

## 1. What is an integral transform?

An integral transform is a mathematical operation that converts a function from one domain to another. It involves integrating the function with a kernel function, resulting in a new function in the transformed domain.

## 2. Why is finding the inverse of an integral transform important?

Finding the inverse of an integral transform allows us to convert a function back to its original domain, making it easier to analyze and understand. It also allows us to solve differential equations and other problems in a simpler form.

## 3. How do you find the inverse of an integral transform?

The process of finding the inverse of an integral transform involves using a specific formula or method, depending on the type of integral transform. This often includes solving a differential equation or using a table of transforms.

## 4. What are some common integral transforms?

Some common integral transforms include the Fourier transform, Laplace transform, and Z-transform. These transforms are often used in engineering, physics, and other scientific fields to convert functions from time or space domains to frequency or complex domains.

## 5. What are some applications of finding the inverse of an integral transform?

Some applications of finding the inverse of an integral transform include solving differential equations, analyzing signals and systems, and solving problems in physics and engineering. It is also used in image and sound processing, data compression, and cryptography.