A hierarchical classification of signals and corresponding approximation method based on minimum norm criterion

This paper presents a hierarchical classification of signals based on their smoothness. By this hierarchical classification, we can obtain the class of bandlimited signals as an innermost signal class and the class of signals composed of differentiable and square integrable functions as the outermost class. Moreover, for each class of signals, we can define "minimum norm signal". The minimum norm signal is defined as the signal of minimum norm which takes specified sample values on a set of given sampling points. By making use of the minimum norm signal, we can construct a unified and efficient approximation method for all these classes of signals. The method has the following special features: i) it is free from numerical integration error, ii) the sequence of approximate signals is guaranteed to uniformly converge to the desired signal as the number of sampling points is increased infinitely.