Ebook: Chebyshev Splines and Kolmogorov Inequalities

This monograph describes advances in the theory of extremal problems in classes of functions defined by a majorizing modulus of continuity w. In particular, an extensive account is given of structural, limiting, and extremal properties of perfect w-splines generalizing standard polynomial perfect splines in the theory of Sobolev classes. In this context special attention is paid to the qualitative description of Chebyshev w-splines and w-polynomials associated with the Kolmogorov problem of n-widths and sharp additive inequalities between the norms of intermediate derivatives in functional classes with a bounding modulus of continuity. Since, as a rule, the techniques of the theory of Sobolev classes are inapplicable in such classes, novel geometrical methods are developed based on entirely new ideas. The book can be used profitably by pure or applied scientists looking for mathematical approaches to the solution of practical problems for which standard methods do not work. The scope of problems treated in the monograph, ranging from the maximization of integral functionals, characterization of the structure of equimeasurable functions, construction of Chebyshev splines through applications of fixed point theorems to the solution of integral equations related to the classical Euler equation, appeals to mathematicians specializing in approximation theory, functional and convex analysis, optimization, topology, and integral equations

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Content:
Front Matter....Pages i-xiii
Introduction....Pages 1-11
Auxiliary Results....Pages 13-18
Maximization of Functionals in H ?[a, b] and Perfect ?-Splines....Pages 19-41
Fredholm Kernels....Pages 43-48
Review of Classical Chebyshev Polynomial Splines....Pages 49-64
Additive Kolmogorov—Landau Inequalities....Pages 65-69
Proof of the Main Result....Pages 71-88
Properties of Chebyshev ?-Splines....Pages 89-100
Chebyshev ?-Splines of the Half-line ?+....Pages 101-109
Maximization of Integral Functionals in H ? [a 1, a 2], - ? ? a 1 < a 2 ? +?....Pages 111-122
Sharp Kolmogorov Inequalities in W r H ?(?)....Pages 123-129
Landau and Hadamard Inequalities in W 1 H ?(?+) and W 1 H ?(?)....Pages 131-138
Sharp Kolmogorov-Landau Inequalities in W 2 H ?(I), I = ? ? ?+....Pages 139-149
Chebyshev ?- Splines and N-Widths of W r H ?[0, 1]....Pages 151-157
Function in W r H ?[-1, 1] Deviating Most from Polynomials $sumlimits_{i = 1}^r {a_i t^i }$ ....Pages 159-164
N-widths of the class W 1 H ?[-1, 1]....Pages 165-178
Lower Bounds for the N-Widths of the Class W r H ?[n]....Pages 179-186
Back Matter....Pages 187-210

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Content:
Front Matter....Pages i-xiii
Introduction....Pages 1-11
Auxiliary Results....Pages 13-18
Maximization of Functionals in H ?[a, b] and Perfect ?-Splines....Pages 19-41
Fredholm Kernels....Pages 43-48
Review of Classical Chebyshev Polynomial Splines....Pages 49-64
Additive Kolmogorov—Landau Inequalities....Pages 65-69
Proof of the Main Result....Pages 71-88
Properties of Chebyshev ?-Splines....Pages 89-100
Chebyshev ?-Splines of the Half-line ?+....Pages 101-109
Maximization of Integral Functionals in H ? [a 1, a 2], - ? ? a 1 < a 2 ? +?....Pages 111-122
Sharp Kolmogorov Inequalities in W r H ?(?)....Pages 123-129
Landau and Hadamard Inequalities in W 1 H ?(?+) and W 1 H ?(?)....Pages 131-138
Sharp Kolmogorov-Landau Inequalities in W 2 H ?(I), I = ? ? ?+....Pages 139-149
Chebyshev ?- Splines and N-Widths of W r H ?[0, 1]....Pages 151-157
Function in W r H ?[-1, 1] Deviating Most from Polynomials $sumlimits_{i = 1}^r {a_i t^i }$ ....Pages 159-164
N-widths of the class W 1 H ?[-1, 1]....Pages 165-178
Lower Bounds for the N-Widths of the Class W r H ?[n]....Pages 179-186
Back Matter....Pages 187-210
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