Area-to-point regression kriging (ATPRK) is an advanced image fusion approach in remote sensing In this paper, ATPRK is considered for sharpening hyperspectral images (HSIs), based on the availability of a fine spatial resolution panchromatic or multispectral image. ATPRK can be used straightforwardly to sharpen each coarse hyperspectral band in turn. This scheme, however, is computationally expensive due to the large number of bands in HSIs, and this problem is exacerbated for multiscene or multitemporal analysis. Thus, we extend ATPRK for fast HSI sharpening with a new approach, called approximate ATPRK (AATPRK), which transforms the original HSI to a new feature space and image fusion is performed for only the first few components before back transformation. Experiments on two HSIs show that AATPRK greatly expedites ATPRK, but inherits the advantages of ATPRK, including maintaining a very similar performance in sharpening (both ATPRK and AATPRK can produce more accurate results than seven benchmark methods) and precisely conserving the spectral properties of coarse HSIs.