Algebraic geometry
For IIR oversampled filter banks, perfect reconstruction have been studied in Wolovich and Kailath. in the context of control theory. While for FIR oversampled filter banks we have to use a different strategy for 1-D and M-D, FIR filter are more popular since they are easier to implement. For 1-D oversampled FIR filter banks, the Euclidean algorithm plays a key role in the matrix inverse problem. However, the Euclidean algorithm fails for multidimensional (MD) filters. For MD filter, we can convert the FIR representation into a polynomial representation. and then use Algebraic geometry and Gröbner bases to get the framework and the reconstruction conditions for the multidimensional oversampled filter banks.