Approximation Theory: Moduli of Continuity and Global Smoothness Preservation

by Anastassiou, George A.; Gal, Sorin G.

• ISBN13: 9780817641511
• ISBN10: 0817641513
• Format: Hardcover
• Copyright: 2000-01-01
• Publisher: Birkhauser

Summary

This monograph, in two parts, is an intensive and comprehensive study of the computational aspects of the moduli of smoothness and the Global Smoothness Preservation Property (GSPP). Key features include: * systematic and extensive study of the computation of Moduli of Continuity and GSPP, presented for the first time in book form * substantial motivation and examples for key results * extensive applications of moduli of smoothness and GSPP concepts to approximation theory, probability theory, numerical and functional analysis * GSPP methods to benefit engineers in computer-aided geometric design * good bibliography and index For researchers and graduate students in pure and applied mathematics.

Table of Contents

Preface	v
Introduction	1	(54)
On Chapter 2: Uniform Moduli of Smoothness	1	(7)
On Chapter 3: Lp-Moduli of Smoothness, 1 ≤ p < ∞	8	(3)
On Chapter 4: Moduli of Smoothness of Special Type	11	(3)
On Chapter 5: Global Smoothness Preservation by Trigonometric Operators	14	(2)
On Chapter 6: Global Smoothness Preservation by Algebric Interpolation Operators	16	(3)
On Chapter 7: Global Smoothness Preservation by General Operators	19	(1)
On Chapter 8: Global Smoothness Preservation by Multivariate Operators	20	(2)
On Chapter 9: Stochastic Global Smoothness Preservation	22	(2)
On Chapter 10: Shift Invariant Univariate Integral Operators	24	(2)
On Chapter 11: Shift Invariant Multivariate Integral Operators	26	(2)
On Chapter 12: Differentiated Shift Invariant Univariate Integral Operators	28	(3)
On Chapter 13: Differentiated Shift Invariant Multivariate Integral Operators	31	(2)
On Chapter 14: Generalized Shift Invariant Univariate Integral Operators	33	(2)
On Chapter 15: Generalized Shift Invariant Multivariate Integral Operators	35	(3)
On Chapter 16: General Theory of Global Smoothness Preservation by Univariate Singular Integrals	38	(3)
On Chapter 17: General Theory of Global Smoothness Preservation by Multivariate Singular Integrals	41	(4)
On Chapter 18: Gonska Progress in Global Smoothness Preservation	45	(1)
On Chapter 19: Miscellaneous Progress on Global Smoothness Preservation	45	(1)
On Chapter 20: Other Applications of the Global Smoothness Preservation Property	45	(1)
Some History of GSPP	46	(4)
Conclusion	50	(5)
Part I Calculus of the Moduli of Smoothness in Classes of Functions	55	(146)
Uniform Moduli of Smoothness	57	(88)
Modulus of Smoothness for Nonperiodic Functions of One Variable	57	(18)
Modulus of Smoothness for Periodic Functions	75	(5)
Bivariate Modulus of Smoothness	80	(9)
Ditzian-Totik Modulus of Smoothness	89	(15)
Applications	104	(39)
Bibliographical Remarks and Open Problems	143	(2)
LP-Moduli of Smoothness, 1 ≤ P < +∞	145	(26)
Usual Lp-Modulus of Smoothness	145	(9)
Averaged Lp-Modulus of Smoothness	154	(9)
Ditzian-Totik Lp-Modulus of Smoothness	163	(2)
Applications	165	(4)
Bibliographical Remarks and Open Problems	169	(2)
Moduli of Smoothness of Special Type	171	(30)
One-Sided Modulus of Smoothness	171	(5)
Hausdorff-Sendov Modulus of Continuity	176	(8)
An Algebraic Modulus of Smoothness	184	(3)
Weighted Moduli of Smoothness	187	(6)
Applications	193	(3)
Bibliographical Remarks and Open Problems	196	(5)
Part II Global Smoothness Preservation by Linear Operators	201	(298)
Global Smoothness Preservation by Trigonometric Operators	203	(8)
General Results	203	(2)
Global Smoothness Preservation by Some Concrete Trigonometric Operators	205	(3)
Global Smoothness Preservation by Trigonometric Projection Operators	208	(2)
Bibliographical Remarks and Open Problems	210	(1)
Global Smoothness Preservation by Algebraic Interpolation Operators	211	(20)
Negative Results	211	(3)
Global Smoothness Preservation by Some Lagrange, Hermite-Fejer and Shepard Operators	214	(10)
Global Smoothness Preservation by Algebraic Projection Operators	224	(3)
Global Smoothness Preservation by Algebraic Polynomials of Best Approximation	227	(3)
Bibliographical Remarks and Open Problems	230	(1)
Global Smoothness Preservation by General Operators	231	(20)
Introduction	231	(2)
General Results	233	(8)
Applications	241	(3)
Variation-Diminishing Splines	241	(2)
Operators of Kratz and Stadtmuller	243	(1)
Optimality of the Preceding Results	244	(7)
Global Smoothness Preservation by Multivariate Operators	251	(14)
Introduction	251	(2)
A General Result for Operators Possessing the Splitting Property	253	(1)
Bernstein Operators over Simplices	254	(2)
Tensor Product Bernstein Operators	256	(2)
An Identity Between K-Functionals and More Results on Global Smoothness	258	(2)
Example: A Comparison Theorem in Stochastic Approximation	260	(5)
Stochastic Global Smoothness Preservation	265	(14)
Introduction	265	(1)
Preliminaries	266	(1)
A Theorem on Stochastic Global Smoothness Preservation	267	(1)
Applications	268	(11)
Stochastic Convolution-Type Operators on C0Ω[a,b]	268	(7)
Operators on CΩ[a,b]	275	(2)
More Convolution-Type Operators	277	(2)
Shift Invariant Univariate Integral Operators	279	(18)
Introduction	279	(2)
General Theory	281	(6)
Applications	287	(10)
Shift Invariant Multivariate Integral Operators	297	(28)
General Results	297	(15)
Applications	312	(13)
Differentiated Shift Invariant Univariate Integral Operators	325	(22)
Introduction	325	(3)
Other Motivations	326	(2)
General Results	328	(5)
Applications	333	(14)
Differentiated Shift Invariant Multivariate Integral Operators	347	(26)
Introduction	347	(3)
General Results	350	(7)
Applications	357	(16)
Generalized Shift Invariant Univariate Integral Operators	373	(18)
General Theory	373	(9)
Applications	382	(9)
Generalized Shift Invariant Multivariate Integral Operators	391	(10)
General Theory	391	(8)
Applications	399	(2)
General Theory of Global Smoothness Preservation by Univariate Singular Operators	401	(28)
Introduction	401	(6)
General Theory	407	(22)
General Theory of Global Smoothness Preservation by Multivariate Singular Operators	429	(22)
Introduction	429	(3)
General Results	432	(19)
Gonska Progress in Global Smoothness Preservation	451	(22)
Simultaneous Global Smoothness Preservation	451	(10)
Bivariate Global Smoothness Preservation by Boolean Sum Operators	461	(4)
Global Smoothness Preservation with Respect to ω2	465	(1)
Global Smoothness Preservation for Bernstein Polynomials Blossoms	466	(2)
Global Smoothness Preservation for Boolean Sums of Convolution Type Operators	468	(5)
Miscellaneous Progress in Global Smoothness Preservation	473	(12)
Preservation of Lipschitz Classes by Bernstein-Type Operators	473	(1)
Preservation of Lipschitz Classes by Some Positive Linear Operators over Unbounded Intervals	474	(4)
Global Smoothness Preservation of Generalized Bernstein-Kantorovich Operators	478	(2)
Global Smoothness Preservation for Generalized Szasz-Kantorovich Operators	480	(2)
First Order Optimal Global Smoothness Preservation for Bernstein-Type Operators	482	(3)
Other Applications of the Global Smoothness Preservation Property	485	(14)
Relationships of the Global Smoothness Preservation Property with the Shape Preservation and the Variation Diminishing Properties	485	(6)
Global Smoothness Preservation in CAGD	491	(4)
Other Applications	495	(2)
Bibliographical Remarks	497	(2)
References	499	(18)
List of Symbols	517	(6)
Index	523