On the Stability of Fourier Phase Retrieval

Phase retrieval is concerned with recovering a function f from the absolute value of its Fourier transform vertical bar(f) over cap vertical bar. We study the stability properties of this problem in Lebesgue spaces. Our main results shows that parallel to f - g parallel to(L2(Rn)) <= 2 center dot parallel to vertical bar(f) over cap vertical bar - vertical bar(g) over cap vertical bar parallel to(L2(Rn)) + h(f) (parallel to f - g parallel to(Lp(Rn))) + J((f) over cap, (g) over cap), where 1 <= p < 2, h(f) is an explicit nonlinear function depending on the smoothness of f and J is an explicit term capturing the invariance under translations. A noteworthy aspect is that the stability is phrased in terms of L-p for 1 <= p < 2: in this region L-p cannot be used to control L-2, our stability estimate has the flavor of an inverse Holder inequality. It seems conceivable that the estimate is optimal up to constants.