Methods of Approximation Theory

Alexander I. Stepanets

• Publisher: De Gruyter
• Year: 2011
• Print ISBN: 9783110630022
• eBook ISBN: 9783110195286
• Edition: 1
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

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Table of contents

• 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