# How to numerically transform a 2D Fourier spectrum with arbitrary frequency shift to center frequency?

Suppose $F(u,v)$ is the center frequency Fourier representation of some $f(x,y)$ in 2D.

$$f(x,y)=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}F(u,v)e^{2\pi i (xu+yv)}dudv$$

In principle for the same $f(x,y)$, the representation could also be $F(u-u_0, v-v_0)$, where $u_0$ and $v_0$ are arbitrarily shifted frequencies.

Now assume the fact is we already have a representation $F(u-u_0, v-v_0)$ and coefficients(arranged as a matrix) and also we know the shift $u_0$ and $v_0$, how can we derive the coefficients of $F(u,v)$ corresponding to the same $f(x,y)$?