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By a technique combining variational methods, contour integration, and the concept of analytic extension, two network theorems are derived which form the basis of a procedure for determining the physically realizable network characteristic which is the best compromise between conflicting requirements in the sense of least mean-square error or some other criterion of approximation selected by the designer and written into the variational problem. Both the specifications and the criterion for compromise may include functions which are given as curves versus frequency rather than explicit functions of frequency. The Wiener-Kolmogoroff theory of smoothing and prediction and the vestigial sideband filter with linear phase shift are included as illustrative application problems. For the former an alternative derivation in the frequency domain as well as a simpler procedure for numerical computations are given. For the latter a criterion for selecting the constant slope of the phase characteristic is derived in addition to providing the procedure for computing the optimum response filter.