Titel: Chebyshev Splines and Kolmogorov Inequalities (Operator Theory: Advances and Applications)
Verlag: Birkh�user
Erscheinungsjahr: 2013
Sprache: Englisch
ISBN-10: 3034897812
ISBN-13: 9783034897815
Einband: Softcover
Zustand: New

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Inhaltsangabe

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 .

Rese�a del editor

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 .

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