A generalized sampling theorem

Using the theory of pseudo‐biorthogonal base and the notion of reproducing kernel, we derive a very broad generalized sampling theorem with real pulse. In addition, it includes the traditional sampling theorem with ideal pulse as its special case. It also includes all the sampling theorems in the cases of bandpass‐type band‐limited signal space for nonuniformly spaced sampling points, for many variables, and for the case in which the notion of frequency is extended from Fourier transform to general integral transform. This generalized sampling theorem is effective also for so‐called undersampling whereby there are too few sampling points compared with the dimension of signal space, and for so‐called oversampling due to too many sampling points. Moreover, it is effective in the case where both occur; that is, in the case where it is undersampling from the standpoint of signal space while it is oversampling in a space of restored signal. For undersampling, the generalized sampling theorem provides the best approximation for each original signal.