Chebyshev Splines and Kolmogorov Inequalities

ISBN-13: 9783034897815 / Angielski / Miękka / 2013 / 210 str.

Sergey Bagdasarov

Opis

Since the introduction of the functional classes HW (lI) and WT HW (lI) and their peri- odic analogs Hw (1I') and (1I'), defined by a concave majorant w of functions and their rth derivatives, many researchers have contributed to the area of ex- tremal problems and approximation of these classes by algebraic or trigonometric polynomials, splines and other finite dimensional subspaces. In many extremal problems in the Sobolev class W (lI) and its periodic ana- log W (1I') an exceptional role belongs to the polynomial perfect splines of degree r, i.e. the functions whose rth derivative takes on the values -1 and 1 on the neighbor- ing intervals. For example, these functions turn out to be extremal in such problems of approximation theory as the best approximation of classes W (lI) and W (1I') by finite-dimensional subspaces and the problem of sharp Kolmogorov inequalities for intermediate derivatives of functions from W . Therefore, no advance in the T exact and complete solution of problems in the nonperiodic classes W HW could be expected without finding analogs of polynomial perfect splines in WT HW .

Spis treści

0 Introduction.- 1 Auxiliary Results.- 2 Maximization of Functionals in H? [a, b] and Perfect ?-Splines.- 3 Fredholm Kernels.- 4 Review of Classical Chebyshev Polynomial Splines.- 5 Additive Kolmogorov-Landau Inequalities.- 6 Proof of the Main Result.- 7 Properties of Chebyshev ?-Splines.- 8 Chebyshev ?-Splines on the Half-line ?+.- 9 Maximization of Integral Functional in H?[a1, a2], -? ? a1 < a2 ? +?.- 10 Sharp Kolmogorov Inequalities in WrH?(?).- 11 Landau and Hadamard Inequalities in WrH?(?+) and WrH?(?).- 12 Sharp Kolmogorov-Landau inequalities in W2H?(?) AND W2H?(?+.- 13 Chebyshev ?-Splines in the Problem of N-Width of the Functional Class WrH?[0, 1].- 14 Function in WrH?[-1, 1] Deviating Most from Polynomials of Degree r.- 15 N-Widths of the Class WrH?[-1, 1].- 16 Lower Bounds for the N-Widths of the Class WrH?[n].- Appendix A Kolmogorov Problem for Functions.- A.3 Sufficient conditions of extremality in the problem (K - L).- A.3.1 Corollaries of differentiation formulas.- A.3.2 Extremality conditions in the form of an operator equation.- A.4.2 Solution of the problem (K).- A.4.3 Problem (K) in the Hölder classes.- B.1 Preliminary remarks.- B.2 Maximization of the norm.- B.2.1 Differentiation formulae and inequalities.- B.3 Maximization of the norm.- B.4 Maximization of the norm.- B.5 Maximization of the norm.