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This paper presents the first results in an unconventional approach to the problem of mean-square optimization. Instead of obtaining a representation for the optimal operator for a process, we seek to characterize the class of processes for which the optimal operator is of specified form. If the processes are given, so that the multivariate characteristic functions are known, then our results can be used to tell whether it is possible for the optimal operator to have a specified form. The bulk of the paper pertains to the signal extraction problem where the signal and noise are independent and additive, and it is desired to estimate some function of the signal. Here, with a slight shift in viewpoint, we phrase the characterization problem in the following way: Given, for example, a noise process, determine the class of signal processes for which the optimal extraction system is of specified form. The case where the noise process is Gaussian comes in for special attention.