Approximation of Functions of Several Variables and Imbedding Theorems: Grundlehren der mathematischen Wissenschaften, cartea 205

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. AüDroximation 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 of0) over Functions of Exponential Type.- 8.8. Decomposition of a Regular Function into Series Relative to de la Vallée Poussin Sums.- 8.9. Representation of Functions of the Classes Bp?r in Terms of de la Vallée Poussin Series. Null Classes (1 ? p ? ?).- 8.10. Series Relative to Dirichlet Sums (1 < p < ?).- 9. The Liouville Classes L.- 9.1. Introduction.- 9.2. Definitions and BasicProperties 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.