✍ Scribed by Alexander I. Stepanets
No coin nor oath required. For personal study only.
PREFACE
PART I
1. REGULARITY OF LINEAR METHODS OF SUMMATION OF FOURIER SERIES
1. Introduction
2. Nikol’skii and Nagy Theorems
3. Lebesgue Constants of Classical Linear Methods
4. Lower Bounds for Lebesgue Constants
5. Linear Methods Determined by Rectangular Matrices
6. Estimates for Integrals of Moduli of Functions Defined by Cosine and Sine Series
7. Asymptotic Equality for Integrals of Moduli of Functions Defined by Trigonometric Series. Telyakovskii Theorem
8. Corollaries of Theorem 7.1. Regularity of Linear Methods of Summation of Fourier Series
2. SATURATION OF LINEAR METHODS
1. Statement of the Problem
2. Sufficient Conditions for Saturation
3. Saturation Classes
4. Criterion for Uniform Boundedness of Multipliers
5. Saturation of Classical Linear Methods
3. CLASSES OF PERIODIC FUNCTIONS
1. Sets of Summable Functions. Moduli of Continuity
2. Classes Hω[a,b] and Hω
3. Moduli of Continuity in Spaces Lp. Classes Hωp
4. Classes of Differentiable Functions
5. Conjugate Functions and Their Classes
6. Weyl-Nagy Classes
7. Classes LψϐN
8. Classes CψϐN
9. Classes LψϐN
10. Order Relation for (ψ,ϐ) -Derivatives
11. ψ-Integrals of Periodic Functions
12. Sets M0, M∞, and Mc
13. Set F
14. Two Counterexamples
15. Function ηa(t) and Sets Defined by It
16. Sets B and M0
4. INTEGRAL REPRESENTATIONS OF DEVIATIONS OF POLYNOMIALS GENERATED BY LINEAR PROCESSES OF SUMMATION OF FOURIER SERIES
1. First Integral Representation
2. Second Integral Representation
3. Representation of Deviations of Fourier Sums on Sets CψM and Lψ
5. APPROXIMATION BY FOURIER SUMS IN SPACES C AND L1
1. Simplest Extremal Problems in Space C
2. Simplest Extremal Problems in Space L1
3. Approximations of Functions of Small Smoothness by Fourier Sums
4. Auxiliary Statements
5. Proofs of Theorems 3.1-3.3'
6. Approximation by Fourier Sums on Classes Hω
7. Approximation by Fourier Sums on Classes Hω
8. Analogs of Theorems 3.1-3.3' in Integral Metric
9. Analogs of Theorems 6.1 and 7.1 in Integral Metric
10. Approximations of Functions of High Smoothness by Fourier Sums in Uniform Metric
11. Auxiliary Statements
12. Proofs of Theorems 10.1-10.3'
13. Analogs of Theorems 10.1-10.3' in Integral Metric
14. Remarks on the Solution of Kolmogorov-Nikol’skii Problem
15. Approximation of ψ-Integrals That Generate Entire Functions by Fourier Sums
16. Approximation of Poisson Integrals by Fourier Sums
17. Corollaries of Telyakovskii Theorem
18. Solution of Kolmogorov-Nikol’skii Problem for Poisson Integrals of Continuous Functions
19. Lebesgue Inequalities for Poisson Integrals
20. Approximation by Fourier Sums on Classes of Analytic Functions
21. Convergence Rate of Group of Deviations
22. Corollaries of Theorems 21.1 and 21.2. Orders of Best Approximations
23. Analogs of Theorems 21.1 and 21.2 and Best Approximations in Integral Metric
24. Strong Summability of Fourier Series
BIBLIOGRAPHICAL NOTES (Part I)
REFERENCES (Part I)
PART II
6. CONVERGENCE RATE OF FOURIER SERIES AND THE BEST APPROXIMATIONS IN THE SPACES Lp
0. Introduction
1. Approximations in the Space L2
2. Direct and Inverse Theorems in the Space L2
3. Extension to the Case of Complete Orthonormal Systems
4. Jackson Inequalities in the Space L2
5. Marcinkiewicz, Riesz, and Hardy-Littlewood Theorems
6. Imbedding Theorems for the Sets LψLP
7. Approximations of Functions from the Sets LψLp by Fourier Sums
8. Best Approximations of Infinitely Differentiable Functions
9. Jackson Inequalities in the Spaces C and Lp
7. BEST APPROXIMATIONS IN THE SPACES C AND L
1. Chebyshev and de la Vallée Poussin Theorems
2. Polynomial of the Best Approximation in the Space L
3. General Facts on the Approximations of Classes of Convolutions
4. Orders of the Best Approximations
5. Exact Values of the Upper Bounds of Best Approximations
6. Dzyadyk-Stechkin-Xiung Yungshen Theorem. Korneichuk Theorem
7. Serdyuk Theorem
8. Bernstein Inequalities for Polynomials
9. Inverse Theorems
8. INTERPOLATION
1. Interpolation Trigonometric Polynomials
2. Lebesgue Constants and Nikol’skii Theorems
3. Approximation by Interpolation Polynomials in the Classes of Infinitely Differentiable Functions
4. Approximation by Interpolation Polynomials on the Classes of Analytic Functions
5. Summable Analog of the Favard Method
9. APPROXIMATIONS IN THE SPACES OF LOCALLY SUMMABLE FUNCTIONS
1. Spaces Lp
2. Order Relation for (ψ, ß)-Derivatives
3. Approximating Functions
4. General Estimates
5. On the Functions ψ(•) Specifying the Sets Lψß
6. Estimates of the Quantities ║ȓcσ(t, ß)║1 for c = σ - h and h > 0
7. Estimates of the Quantities ║ȓcσ(t, ß)║1 for c = θσ, 0 ≤θ≤ 1, and ψ ∈ Uc
8. Estimates of the Quantities ║ȓcσ(t, ß)║1 for c = 2σ - η(σ) and ψ ∈ U∞
9. Estimates of the Quantities ║ȓcσ(t, 0)║1 for c = θσ, 0 ≤ θ ≤ 1, and ψ ∈ U0
10. Estimates of the Quantities ║δσ,c(t,ß)║1
11. Basic Results
12. Upper Bounds of the Deviations ρσ(f;•) in the Classes Ĉψß,∞ and ĈψßHω
13. Some Remarks on the Approximation of Functions of High Smoothness
14. Strong Means of Deviations of the Operators Fσ(f;x)
10. APPROXIMATION OF CAUCHY-TYPE INTEGRALS
1. Definitions and Auxiliary Statements
2. Sets of ψ-Integrals
3. Approximation of Functions from the Classes Cψ(T)+
4. Landau Constants
5. Asymptotic Equalities
6. Lebesgue-Landau Inequalities
7. Approximation of Cauchy-Type Integrals
11. APPROXIMATIONS IN THE SPACES SP
1. Spaces
2. ψ-Integrals and Characteristic Sequences
3. Best Approximations and Widths of p-Ellipsoids
4. Approximations of Individual Elements from the Sets
5. Best n-Term Approximations
6. Best n-Term Approximations (q>p)
7. Proof of Lemma 6.1
8. Best Approximations by q-Ellipsoids in the Spaces Spφ
9. Application of Obtained Results to Problems of Approximation of Periodic Functions of Many Variables
10. Remarks
11. Theorems of Jackson and Bernstein in the Spaces Sp
12. APPROXIMATIONS BY ZYGMUND AND DE LA VALLÉE POUSSIN SUMS
1. Fejér Sums: Survey of Known Results
2. Riesz Sums: A Survey of Available Results
3. Zygmund Sums: A Survey of Available Results
4. Zygmund Sums on the Classes Cψß,∞
5. De la Vallée Poussin Sums on the Classes Wrß and WrßHw
6. De la Vallée Poussin Sums on the Classes CψßN and CψN
BIBLIOGRAPHICAL NOTES (Part II)
REFERENCES (Part II)
Index
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