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.