Approximation of Functions of Several Variables and Imbedding Theorems (Grundlehren der mathematischen Wissenschaften, 205)

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 Front Matter....Pages I-VIII Introduction....Pages 1-4 Preparatory Information....Pages 5-80 Trigonometric Polynomials....Pages 81-97 Entire Functions of Exponential Type, Bounded on R....Pages 98-140 The Function Classes W , H , B ....Pages 141-182 Direct and Inverse Theorems of the Theory of Approximation. Equivalent Norms....Pages 183-230 Imbedding Theorems for Different Metrics and Dimensions....Pages 231-260 Transitivity and Unimprovability of Imbedding Theorems. Compactness....Pages 261-288 Integral Representations and Isomorphism of Isotropy Classes....Pages 289-322 The Liouville Classes L ....Pages 323-376 Back Matter....Pages 377-420