An Extension of Oblique Projection Sampling Theorem

In this paper, we discuss the sampling problem without a condition that was assumed in conventional sampling theorems. This means that we cannot perfectly reconstruct all functions in the reconstruction space. The perfect reconstruction is possible only for functions in an arbitrary complementary subspace of the intersection of the reconstruction space and the orthogonal complement of the sampling space. We propose a sampling theorem that reconstructs the oblique projection onto the complementary subspace along the orthogonal complement of the sampling space. The sampling theorem guarantees the perfect reconstruction of functions of special interest in the reconstruction space, such as the constant function in image processing applications. In addition, we explain why a conventional sampling theorem is not suitable for the present case.