Moduli of Smoothness

I. The Modulus of Smoothness.- 1. Preliminaries.- 1.1. Notations.- 1.2. Discussion of Some Conditions on ?(x).- 1.3. Examples of Various Step-Weight Functions ?(x).- 2. The K-Functional and the Modulus of Continuity.- 2.1. The Equivalence Theorem.- 2.2. The Upper Estimate, Kr, ?(f,tr)p ? M??r(f,t)p, Case I.- 2.3. The Upper Estimate of the K-Functional, The Other Cases.- 2.4. The Lower Estimate for the K-Functional.- 3. K-Functionals and Moduli of Smoothness, Other Forms.- 3.1. A Modified K-Functional.- 3.2. Forward and Backward Differences.- 3.3. Main-Part Modulus of Smoothness.- 3.4. Computation of Our Modulus for Some Functions.- 4. Properties of ??r(f,t)p.- 4.1. Extending the Basic Properties of the Classical Moduli.- 4.2. Optimal Rate of ??r(f,t).- 4.3. Marchaud Inequality.- 5. More General Step-Weight Functions ?.- 5.1. Logarithmic-Type Weights and Internal Zeros.- 5.2. The Necessity of the Finite Overlapping Condition.- 5.3. Growth Order of Type x? with Arbitrary ?.- 6. Weighted Moduli of Smoothness.- 6.1. Weighted Moduli of Smoothness and Weighted K-Functionals.- 6.2. The Weighted Main-Part Modulus.- 6.3. Smoothness Properties of Derivatives.- 6.4. Marchaud Inequality for Weighted Main-Part Moduli.- 6.5. Connection with Ordinary Weighted Moduli.- II. Applications.- 7. Algebraic Polynomial Approximation.- 7.1. Background.- 7.2. Best Polynomial Approximation.- 7.3. Asymptotic Behavior of Derivatives of Best Approximating Polynomials.- 7.4. Error Bounds for Gaussian Quadrature.- 8. Weighted Best Polynomial Approximation.- 8.1. Some Concepts and Description of the Weight.- 8.2. Best Weighted Algebraic Polynomial Approximation.- 8.3. Derivatives of the Optimal Polynomials.- 8.4. Proof of Some Crucial Inequalities for w ? Jp*.- 8.5. Applications, Calculations, and Specific Examples.- 9. Exponential-Type or Bernstein-Type Operators.- 9.1. Background and Notations, Positive Operators on C(D).- 9.2. Operators on Lp(D), Higher Degree of Smoothness.- 9.3. Direct and Converse Results.- 9.4. The Bernstein-Type Inequality ??2rLn(2r)f?p ? Mnr ?f?p.- 9.5. Rate of Convergence for Smooth Functions.- 9.6. Estimate of ?Ln(R2r(f, · ,x), x)?Lp(En).- 9.7. The Estimate ??(x)2rLn(2r)(f)?Lp ? M??2rf(2r)?p.- 10. Weighted Approximations by Exponential-Type Operators.- 10.1. The Direct and Inverse Result.- 10.2. The Boundedness of the Operators in Weighted Norm.- 10.3. Bernstein-Type Inequality.- 10.4. The Estimate ?w?2Ln(2)(f)? ? C(?w?2f(2)? + ?f?).- 10.5. The Estimate of Lnf - f for Smooth Functions.- 10.6. The Saturation Result.- 11. Weighted Polynomial Approximation in LP(R).- 11.1. Introduction.- 11.2. The Equivalence Result.- 11.3. The Direct and Converse Results.- 11.4. Proof of the Equivalence Result.- 11.5. Comparisons and Generalizations.- 12. Polynomial Approximation in Several Variables.- 12.1. Approximation on Cubes.- 12.2. Approximation on Polytopes.- 13. Comparisons and Conclusions.- 13.1. Comparison with Similar Expressions.- 13.2. The Integral Modulus of Smoothness of Ivanov and Sendov.- 13.3. Moduli Generated by Multipliers and Integral Transforms.- 13.4. A Modulus Introduced by Potapov.- 13.5. Hoeffding's Result.- 13.6. Conclusion.- A. The Analogue of Definition 5.3.1.- B. The Definition of the Weighted Modulus of Smoothness on (0,1).- References.- List of Symbols.