Classification and Approximation of Periodic Functions

1. Classes of Periodic Functions.- 1. Sets of Summable Functions. Moduli of Continuity.- 2. The Classes H?[a, b] and H?.- 3. Moduli of Continuity in the Spaces Lp. The Classes H?p.- 4. Classes of Differentiable Functions.- 5. Conjugate Functions and Their Classes.- 6. Weil-Nagy Classes.- 7. The Classes.- 8. The Classes.- 9. The Classes 35 10. Order Relation for (?, ? )-Derivatives.- 2. Integral Representations of Deviations of Linear Means Of Fourier Series.- 1. Fourier Sums.- 2. Linear Methods of Summation of Fourier Series. General Aspects.- 3. Integral Representations of ?n(f;x;?).- 4. Representations of Deviations of Fourier Sums on the Sets and.- 5. Representations of Deviations of Fourier Sums on the Sets and.- 3. Approximations by Fourier Sums in the Spaces c and l1.- 1. Simplest Extremal Problems in the Space C.- 2. Simplest Extremal Problems in the Space L1.- 3. Asymptotic Equalities for ? n(H?).- 4. Asymptotic Equalities for.- 5. Moduli of Half-Decay of Convex Functions.- 6. Asymptotic Representations for ?n(f; x) on the Sets.- 7. Asymptotic Equalities for and.- 8. Approximations of Analytic Functions by Fourier Sums in the Uniform Metric.- 9. Approximations of Entire Functions by Fourier Sums in the Uniform Metric.- 10. Asymptotic Equalities for and.- 11. Asymptotic Equalities for and.- 12. Asymptotic Equalities for and.- 13. Approximations of Analytic Functions in the Metric of the Space L.- 14. Asymptotic Equalities for and.- 15. Behavior of a Sequence of Partial Fourier Sums near Their Points of Divergence.- 4. Simultaneous Approximation of Functions and their Derivatives by Fourier Sums.- 1. Statement of the Problem and Auxiliary Facts.- 2. Asymptotic Equalities for.- 3. Asymptotic Equalities for.- 4. Corollaries of Theorems 2.1 and 3.1.- 5.Convergence Rate of the Group of Deviations.- 6. Strong Summability of Fourier Series.- 5. Convergence Rate of Fourier Series and Best Approximations in the Spaces lp.- 1. Approximations in the Space L2.- 2. Jackson Inequalities in the Space L2.- 3. Multiplicators. Marcinkiewicz Theorem. Riesz Theorem. Hardy — Littlewood Theorem.- 4. Imbedding Theorems for the Sets.- 5. Approximations of Functions from the Sets.- 6. Best Approximations of Infinitely Differentiable Functions.- 7. Jackson Inequalities in the Spaces C and Lp.- 6. Best Approximations in the Spaces C and l.- 1. Zeros of Trigonometric Polynomials.- 2. Chebyshev Theorem and de la Vallée Poussin Theorem.- 3. Polynomial of Best Approximation in the Space L.- 4. Approximation of Classes of Convolutions.- 5. Orders of Best Approximations.- 6. Exact Values of Upper Bounds of Best Approximations.- Bibliographical Notes.- References.