  This note complements the paper "The quest for optimal sampling:
Computationally efficient, structure-exploiting measurements for compressed
sensing" [2]. Its purpose is to present a proof of a result stated therein
concerning the recovery via compressed sensing of a signal that has structured
sparsity in a Haar wavelet basis when sampled using a multilevel-subsampled
discrete Fourier transform. In doing so, it provides a simple exposition of the
proof in the case of Haar wavelets and discrete Fourier samples of more general
result recently provided in the paper "Breaking the coherence barrier: A new
theory for compressed sensing" [1].
