How can we solve this integral equation for f(t) given g(x) is known?

[Karlisbad]

i would need to solve the integral equation..

$$g(x)=\int_{-\infty}^{\infty}dte^{ixf(t)}$$

the unknown function is f(t) g(x) is known, appart from using a numerical cuadrature method for the integral, what else can i do?.. could we prove if the function f(t) will exist or if it won't have any solution??..thanx.

 

[DeadWolfe]

"the unknown function is f(t) g(x) is known"

*head explodes*

 

[tacman]

Solving integral equations can be a challenging task, but there are a few different approaches you can take to solve this particular equation. One option is to use a numerical quadrature method, as you mentioned. This involves approximating the integral using discrete points and then solving for the unknown function f(t) at each point. Another option is to use Fourier analysis techniques to transform the integral equation into a differential equation, which may be easier to solve.

In terms of proving the existence of a solution for f(t), it would depend on the specific properties of the function g(x). If g(x) is a well-behaved function, then it is likely that f(t) will also have a solution. However, if g(x) is highly oscillatory or has other complicated properties, it may be more difficult to prove the existence of a solution for f(t).

Overall, it may be helpful to consult with a mathematician or use software specifically designed for solving integral equations to find the best approach for your particular problem. Good luck with solving the equation!