Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets

In this paper we study the general reconstruction of a compactly supported function from its Fourier coefficients using compactly supported shearlet systems. We assume that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements an approximation is sought in a finite dimensional shearlet reconstruction space. We analyze this sampling and reconstruction process by a recently introduced method called generalized sampling. In particular by studying the stable sampling rate of generalized sampling we then show stable recovery of the signal is possible using an almost linear rate. Furthermore, we compare the result to the previously obtained rates for wavelets.