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Table of Contents
Cover
Title page
Basic notation
Chapter 1. Introduction
1.1. Background
1.2. Ulyanov inequalities for moduli of smoothness
1.3. Specifics of the case 0<
<
1
1.4. Moduli of smoothness, -functionals, and their realizations
1.5. Main tool: Hardy-Littlewood-Nikol'skii polynomial inequalities
1.6. Main goals and work organization
Chapter 2. Auxiliary results
2.1. Besov and Triebel-Lizorkin spaces and their embeddings
2.2. Fourier multipliers
2.3. Estimates of _{ }-norm of an integrable function in terms of its Fourier coefficients
2.4. Estimates of _{ }-norms of polynomials related to Lebesgue constants
2.5. Hardy-Littlewood type theorems for Fourier series with monotone coefficients
2.6. Weighted Hardy's inequality for averages
Chapter 3. Polynomial inequalities of Nikol'skii-Stechkin-Boas-types
Chapter 4. Basic properties of fractional moduli of smoothness
Chapter 5. Hardy-Littlewood-Nikol'skii inequalities for trigonometric polynomials
5.1. Hardy-Littlewood-Nikol'skii ( _{ }, _{ }) inequalities for 0<
≤1
5.2. Hardy-Littlewood-Nikol'skii ( _{ }, _{ }) inequalities for 1<
<
≤∞
5.3. Hardy-Littlewood-Nikol'skii ( _{ }, _{ }) inequalities for directional derivatives
Chapter 6. General form of the Ulyanov inequality for moduli of smoothness, -functionals, and their realizations
Chapter 7. Sharp Ulyanov inequalities for -functionals and realizations
Chapter 8. Sharp Ulyanov inequalities for moduli of smoothness
8.1. One-dimensional results
8.2. Comparison between the sharp and classical Ulyanov inequalities
8.3. Multidimensional results
Chapter 9. Sharp Ulyanov inequalities for realizations of -functionals related to Riesz derivatives and corresponding moduli of smoothness
Chapter 10. Sharp Ulyanov inequality via Marchaud inequality.
10.1. Inequalities for realizations of -functionals
10.2. Inequalities for moduli of smoothness
Chapter 11. Sharp Ulyanov and Kolyada inequalities in Hardy spaces
11.1. Sharp Ulyanov and Kolyada inequalities in _{ }(\R^{ })
11.2. Sharp Ulyanov and Kolyada inequalities in analytic Hardy spaces
Chapter 12. ( _{ }, _{ }) inequalities of Ulyanov-type involving derivatives
Chapter 13. Embedding theorems for function spaces
13.1. New embedding theorems for Lipschitz spaces
13.2. Embedding properties of Besov and Lipschitz-type spaces
Bibliography
List of symbols
Back Cover.
Title page
Basic notation
Chapter 1. Introduction
1.1. Background
1.2. Ulyanov inequalities for moduli of smoothness
1.3. Specifics of the case 0<
<
1
1.4. Moduli of smoothness, -functionals, and their realizations
1.5. Main tool: Hardy-Littlewood-Nikol'skii polynomial inequalities
1.6. Main goals and work organization
Chapter 2. Auxiliary results
2.1. Besov and Triebel-Lizorkin spaces and their embeddings
2.2. Fourier multipliers
2.3. Estimates of _{ }-norm of an integrable function in terms of its Fourier coefficients
2.4. Estimates of _{ }-norms of polynomials related to Lebesgue constants
2.5. Hardy-Littlewood type theorems for Fourier series with monotone coefficients
2.6. Weighted Hardy's inequality for averages
Chapter 3. Polynomial inequalities of Nikol'skii-Stechkin-Boas-types
Chapter 4. Basic properties of fractional moduli of smoothness
Chapter 5. Hardy-Littlewood-Nikol'skii inequalities for trigonometric polynomials
5.1. Hardy-Littlewood-Nikol'skii ( _{ }, _{ }) inequalities for 0<
≤1
5.2. Hardy-Littlewood-Nikol'skii ( _{ }, _{ }) inequalities for 1<
<
≤∞
5.3. Hardy-Littlewood-Nikol'skii ( _{ }, _{ }) inequalities for directional derivatives
Chapter 6. General form of the Ulyanov inequality for moduli of smoothness, -functionals, and their realizations
Chapter 7. Sharp Ulyanov inequalities for -functionals and realizations
Chapter 8. Sharp Ulyanov inequalities for moduli of smoothness
8.1. One-dimensional results
8.2. Comparison between the sharp and classical Ulyanov inequalities
8.3. Multidimensional results
Chapter 9. Sharp Ulyanov inequalities for realizations of -functionals related to Riesz derivatives and corresponding moduli of smoothness
Chapter 10. Sharp Ulyanov inequality via Marchaud inequality.
10.1. Inequalities for realizations of -functionals
10.2. Inequalities for moduli of smoothness
Chapter 11. Sharp Ulyanov and Kolyada inequalities in Hardy spaces
11.1. Sharp Ulyanov and Kolyada inequalities in _{ }(\R^{ })
11.2. Sharp Ulyanov and Kolyada inequalities in analytic Hardy spaces
Chapter 12. ( _{ }, _{ }) inequalities of Ulyanov-type involving derivatives
Chapter 13. Embedding theorems for function spaces
13.1. New embedding theorems for Lipschitz spaces
13.2. Embedding properties of Besov and Lipschitz-type spaces
Bibliography
List of symbols
Back Cover.