This paper introduces a novel technique to preserve spectral features in lossy compression based on a novel fast Fourier correction algorithm\added{ for regular-grid data}. Preserving both spatial and frequency representations of data is crucial for applications such as cosmology, turbulent combustion, and X-ray diffraction, where spatial and frequency views provide complementary scientific insights. In particular, many analysis tasks rely on frequency-domain representations to capture key features, including the power spectrum of cosmology simulations, the turbulent energy spectrum in combustion, and diffraction patterns in reciprocal space for ptychography. However, existing compression methods guarantee accuracy only in the spatial domain while disregarding the frequency domain. To address this limitation, we propose an algorithm that corrects the errors produced by off-the-shelf ``base'' compressors such as SZ3, ZFP, and SPERR, thereby preserving both spatial and frequency representations by bounding errors in both domains. By expressing frequency-domain errors as linear combinations of spatial-domain errors, we derive a region that jointly bounds errors in both domains. Given as input the spatial errors from a base compressor and user-defined error bounds in the spatial and frequency domains, we iteratively project the spatial error vector onto the regions defined by the spatial and frequency constraints until it lies within their intersection. We further accelerate the algorithm using GPU parallelism to achieve practical performance. We validate our approach with datasets from cosmology simulations, X-ray diffraction, combustion simulation, and electroencephalography demonstrating its effectiveness in preserving critical scientific information in both spatial and frequency domains.