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 .

Inhalt
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.