Thread: fourier transform

Show that following function is a covariance function by taking the fourier transform.

C( τ) = σ²exp {-a |τ |}(1+a|τ |+((aτ)²)/3),a>=0

I have done the fourier transform before but not sure how to integrate with the absolute value.So an explained solution would be very helpful.

I think you want to use Bochner's theorem, in the context of Probability theory: If the Fourier transform of the function C is a positive-definite complex-valued function, then C is a covariance function.