Moduli of Smoothness (Springer Series in Computational Mathematics)

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9781461291510: Moduli of Smoothness (Springer Series in Computational Mathematics)

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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The book introduces a new way of measuring smoothness. The need for this new concept arises from the failure of the classical moduli of smoothness to solve some basic problems, such as characterizing the behaviour of best polynomial approximation in Lp -1,1 . The new modulus, which has a simple form, can also be described as a Peetre K functional between an Lp space and a weighted Sobolev space. Connections between interpolation of spaces and approximation theory are utilized in applying the modulus of smoothness. The applications include best (weighted) polynomial approximation on a finite interval, characterization of the rate of approximation given by classical operator processes such as Bernstein, Kantorovich, Szasz-Mirakjan, and Post-Widder operators, Freud-type weighted polynomial approximation on infinite intervals with exponentially decreasing weights and polynomial approximation in several variables. Special emphasis is placed on the computability aspect of the moduli. The results are new, and complete proofs are given. It is hoped that the book will be of interest and useful for mathematicians working in approximation theory, interpolation of spaces, numerical analysis and real analysis.

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Z. Ditzian, Vilmos Totik
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Book Description Springer-Verlag New York Inc., United States, 2011. Paperback. Book Condition: New. 229 x 157 mm. Language: English . Brand New Book ***** Print on Demand *****.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 . Softcover reprint of the original 1st ed. 1987. Bookseller Inventory # AAV9781461291510

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Z. Ditzian, Vilmos Totik
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Book Description Springer-Verlag New York Inc., United States, 2011. Paperback. Book Condition: New. 229 x 157 mm. Language: English . Brand New Book ***** Print on Demand *****. 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 . Softcover reprint of the original 1st ed. 1987. Bookseller Inventory # AAV9781461291510

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Book Description Springer, 2011. Paperback. Book Condition: NEW. 9781461291510 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Bookseller Inventory # HTANDREE0301191

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Book Description Springer. Paperback. Book Condition: New. Paperback. 227 pages. Dimensions: 9.0in. x 6.2in. x 0.6in.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 . . . . . . . This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Paperback. Bookseller Inventory # 9781461291510

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Book Description Springer New York, 2011. Paperback. Book Condition: Brand New. reprint edition. 236 pages. 8.75x6.00x0.50 inches. In Stock. Bookseller Inventory # x-1461291518

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