Fourier Transform: Show g(hat)(xi) = e-i*xi.a * f(hat)(xi)

[squenshl]

Homework Statement

Suppose that f has Fourier transform f(hat). If a is a member of Rⁿ, let g be the function defined by g(x) = f(x-a). Show that g(hat)(xi) = e^-i*xi.a * f(hat)(xi).

Homework Equations

The Attempt at a Solution

Is it using the convolution theroem otherwise I am lost.

 

[Dick]

Write down your definition of Fourier transform. Then just apply the change of variable x-a=u. Nothing to do with convolution.

 

[squenshl]

Thanks. Now I'm looking to solve u_t + v . div(u) = D*laplacian(u), x is a member of Rⁿ, t > 0,
u(x,0) = 0, x is a member of Rⁿ, (This is of course a Cauchy problem) where v is a member of Rⁿ and is the constant velocity vector of the medium. D is constant diffusivity.
I can find the Fourier transform of D*laplacian(u) which is -|xi|²*u(hat)(xi) and the Fourier transform of u_t is u(hat)(xi) but how do I find the Fourier transform of the other part. I think we use the Fourier transform of g(hat)(xi) but I'm not sure.

 

[squenshl]

I still don't even know where to start on the final integral.

 

[squenshl]

I know that the Fourier transform of grad(u) = i*u(hat)(xi)*(xi₁+xi₂+...+xi_n) and v is a constant vector so the Fourier transform of v.grad(u) = v.(i*u(hat)(xi)*(xi₁+xi₂+...+xi_n), is there anyway I can simplify this.