I have to solve an exercise which looks like a more general form of Fourier inversion formula. But, I'm having hard time attacking it...
A given function f is integrable on the real line and continuous and f(x) goes to zero as the absolute value of x goes to infinity.
And f satisfies this additional condition.
Here, A is some fixed positive number.
Then, I have to show that the following holds for any real number x.
It seems a lot more complicated than I thought, because f^ doesn't have to be an integrable function. So I'm totally stuck. Could anyone please help me with this exercise? I am quite desperate.