Moduli of Smoothness
Published on 19. October 1987
Book
Hardback
IX, 227 pages
978-0-387-96536-9 (ISBN)
Description
The subject of this book is the introduction and application of a new measure for smoothness offunctions. Though we have both previously published some articles in this direction, the results given here are new. Much of the work was done in the summer of 1984 in Edmonton when we consolidated earlier ideas and worked out most of the details of the text. It took another year and a half to improve and polish many of the theorems. We express our gratitude to Paul Nevai and Richard Varga for their encouragement. We thank NSERC of Canada for its valuable support. We also thank Christine Fischer and Laura Heiland for their careful typing of our manuscript. z. Ditzian V. Totik CONTENTS Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 PART I. THE MODULUS OF SMOOTHNESS Chapter 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1. Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2. Discussion of Some Conditions on cp(x). . . . . . . . . . . . . . . . 8 . . . . 1.3. Examples of Various Step-Weight Functions cp(x) . . . . . . . . . . . 9 . . Chapter 2. The K-Functional and the Modulus of Continuity ... . ... 10 2.1. The Equivalence Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 . . . . . . . . . 2.2. The Upper Estimate, Kr.tp(f, tr)p ~ Mw;(f, t)p, Case I . . . . . . . . . . . . 12 . . . 2.3. The Upper Estimate of the K-Functional, The Other Cases. . . . . . . . . . 16 . 2.4. The Lower Estimate for the K-Functional. . . . . . . . . . . . . . . . . . . 20 . . . . . Chapter 3. K-Functionals and Moduli of Smoothness, Other Forms. 24 3.1. A Modified K-Functional . . . . . . . . . . . . . . . . . . . . . . . . . . 24 . . . . . . . . . . 3.2. Forward and Backward Differences. . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . 3.3. Main-Part Modulus of Smoothness. . . . . . . . . . . . . . . . . . . . . . 28 . . . . . . .
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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, ยท ,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.