This English translation of my book "PribliZenie Funkcir Mnogih Peremennyh i Teoremy Vlozel1iya" is identical in content with the Rus- sian original, published by "Nauka" in 1969. However, I have corrected a number of errors. I am grateful to the publishing house Springer-Verlag for making my book available to mathematicians who do not know Russian. I am also especially grateful to the translator, Professor John M. Dan- skin, who has fulfilled his task with painstaking care. In doing so he has showed high qualifications both as a mathematician and as a translator of Russian, which is considered by many to be a very difficult language. The discussion in this book is restricted, for the most part, to func- tions everywhere defined in n-dimensional space. The study of these questions for functions given on bounded regions requires new methods. In. connection with this I note that a new book, "Integral Represen- tations of Functions and Imbedding Theorems", by O. V. Besov, V. P. Il'in, and myself, has just (May 1975) been published, by the publishing house "Nauka", in Moscow. Moscow, U.S.S.R., May 1975 S. M. Nikol'skir Translator's Note I am very grateful to Professor Nikol'skir, whose knowledge of English, which is considered by many to be a very difficult language, is excellent, for much help in achieving a correct translation of his book. And I join Professor Nikol'skir in thanking Springer-Verlag. The editing problem was considerable, and the typographical problem formidable.

Table of Contents:
1. Preparatory Information.- 1.1. The Spaces C(?) and Lp(?).- 1.2. Normed Linear Spaces.- 1.3. Properties of the Space Lp(?).- 1.4. Averaging of Functions According to Sobolev.- 1.5. Generalized Functions.- 2. Trigonometric Polynomials.- 2.1. Theorems on Zeros. Linear Independence.- 2.2. Important Examples of Trigonometric Polynomials.- 2.3. The Trigonometric Interpolation Polynomial of Lagrange.- 2.4. The Interpolation Formula of M. Riesz.- 2.5. The Bernstein's Inequality.- 2.6. Trigonometric Polynomials of Several Variables.- 2.7. Trigonometric Polynomials Relative to Certain Variables.- 3. Entire Functions of Exponential Type, Bounded on ?n.- 3.1. Preparatory Material.- 3.2. Interpolation Formula.- 3.3. Inequalities of Different Metrics for Entire Functions of Exponential Type.- 3.4. Inequalities of Different Dimensions for Entire Functions of Exponential Type.- 3.5. Subspaces of Functions of Given Exponential Type.- 3.6. Convolutions with Entire Functions of Exponential Type.- 4. The Function Classes W, H, B.- 4.1. The Generalized Derivative.- 4.2. Finite Differences and Moduli of Continuity.- 4.3. The Classes W, H, B.- 4.4. Representation of an Intermediate Derivate in Terms of a Derivative of Higher Order and the Function. Corollaries.- 4.5. More on Sobolev Averages.- 4.6. Estimate of the Increment Relative to a Direction.- 4.7. Completeness of the Spaces W, H, B.- 4.8. Estimates of the Derivative by the Difference Quotient.- 5. Direct and Inverse Theorems of the Theory of Approximation. Equivalent Norms.- 5.1. Introduction.- 5.2. AuDroximation Theorem.- 5.3. Periodic Classes.- 5.4. Inverse Theorems of the Theory of Approximations.- 5.5. Direct and Inverse Theorems on Best Approximations. Equivalent H-Norms.- 5.6. Definition of B-Classes with the Aid of 0) over Functions of Exponential Type.- 8.8. Decomposition of a Regular Function into Series Relative to de la Vallee Poussin Sums.- 8.9. Representation of Functions of the Classes Bp?r in Terms of de la Vallee Poussin Series. Null Classes (1 ? p ? ?).- 8.10. Series Relative to Dirichlet Sums (1 ?).- 9. The Liouville Classes L.- 9.1. Introduction.- 9.2. Definitions and Basic Properties of the Classes Lpr and pr.- 9.3. Interrelationships among Liouville and other Classes.- 9.4. Integral Representation of Anisotropic Classes.- 9.5. Imbedding Theorems.- 9.6. Imbedding Theorem with a Limiting Exponent.- 9.7. Nonequivalence of the Classes Bpr and Lpr.- Remarks.- Literature.- Index of Names.