Time-frequency concentrated basis functions

The notion of concentration is discussed, and concentration with respect to an operator is defined. This is related to the idea of concentration based on second moments in time and frequency. A class of signals B_w(μ) concentrated with respect to an operator W is introduced. A best n-dimensional approximating subspace for this class is shown to be the space spanned by the n signals most concentrated with respect to the operator defining the class. The case of approximating B_W(μ) with a subspace which is not maximally concentrated with respect to W, but which consists of signals which are maximally concentrated with respect to some other operator, is examined. This is motivated from time-frequency analysis where useful physical insight may be gained by decomposing a signal into elementary locally concentrated components. The tradeoff between error and concentration is discussed