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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