Absract.A concise mathematical statement of the signal classification problem is given. Within its framework, linear separability criteria are described for a set of points in a finite-dimensional space and then are generalized to an infinite-dimensional space. The problem of synthesis of both linear and nonlinear classification systems is also solved, for which an algorithm is formulated that chooses optimal structures of sequential and parallel functional transformations of uninformative coordinates in a feature vector.Keywords: clustering, synthesis of classification systems, pattern recognition, conditions of linear and nonlinear separability and divisibility of sets, pseudoinverse and projection matrices.

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