In this paper, we present a novel technique for signal synthesis in the presence of an inconsistent set of constraints. This technique represents a general, minimum norm, solution to the class of synthesis problems in which: the desired signal may be characterized as being an element of some Hilbert Space; each of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> design constraints generates a closed convex set in that space; and those <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> convex sets generate, or may be resolved into, two disjoint closed convex sets, such that at least one of the two sets is bounded. The synthesis technique employs alternating nearest point maps onto closed convex subsets of a Hilbert Space, and may be viewed as an extension of D. Youla's "Method of Convex Projections"--which addresses the case in which the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> closed convex sets, corresponding to the design constraints, possess a nonempty intersection. Section I provides a general introduction to the synthesis problem and to its solution. Section II contains the mathematical justification for the solution technique, while Section III presents an example of the synthesis of a data window for spectral estimation. In Section IV, we discuss potential extensions of this technique within the area of signal synthesis, as well as to the more general class of constrained optimization problems.