Generalized analysis of optimum monotonic low-pass filters

New classes of low-pass filters with no finite zeros having a monotonic passband response are introduced by maximizing or minimizing the mean square error and changing the boundaries of the error integral. It is shown that all known classes of filters with monotonic passband magnitude response (Butterworth, Legendre, LSM, and Halpern filters) are obtained as special cases of this type of approximation.