Linear Superpositions of Functions

CONTENTS Preface Chapter I. Survey of the results § 1. Superpositions of analytic functions § 2. The problem of resolvents § 3. Superpositions of smooth functions and the theory of approximations § 4. Superpositions of continuous functions § 5. Linear superpositions Chapter II. Completeness of the space of linear superpositions § 1. Notation § 2. Estimate of the difference of the integrals of one term of a superposition along nearby level curves § 3. Deletion of dependent terms § 4. Reduction of linear superpositions to a form with independent terms § 5. The set of linear superpositions in the space of continuous functions is closed § 6. The set of linear superpositions in the space of continuous functions is nowhere dense § 7. Kolmogorov' s theorem on superpositions of continuous functions Chapter III. Functional "dimension" of the space of linear superpositions § 1. (ε,δ)-entropy and the "dimension" of function spaces § 2. (ε,δ)-entropy of the set of linear superpositions § 3. Functional "dimension" of the space of linear superpositions § 4. Variation of superpositions of smooth functions § 5. Instability of the representation of functions as superpositions of smooth functions § 6. Mapping of a k-dimensional cube onto an n-dimensional cube § 7. Isomorphic embedding (in the uniform metric) of the space of p-smooth functions of n-variables into the space of sufficiently smooth functions of a smaller number of variables References