Kolmogorov problem in $W^rH^ω[0,1]$ and extremal Zolotarev ω-splines

Bagdasarov Sergey K.

GDML_Books, (1998), p.

Résumé

AbstractThe main result of the paper, based on the Borsuk Antipodality Theorem, describes extremal functions of the Kolmogorov-Landau problem(*) $f^{(m)}(ξ) → sup$, $f ∈ W^rH^ω[ξ,b]$, $||f||_{ℂ[a,b]} ≤ B$,for all 0 < m ≤ r, ξ ≤ a or ξ = (a+b)/2, all B > 0 and concave moduli of continuity ω on ℝ₊. It is shown that any extremal function $ = _{B,r,m,ω,ξ}$ of the problem (*) enjoys the following two characteristic properties. First, the function $^{(r)}(·) - ^{(r)}(ξ)$ is extremal for the problem(**) $∫_ξ^b h(t)ψ(t)dt → sup$, $h ∈ H^ω[ξ,b]$, h(ξ) = 0,for an appropriate choice of the kernel ψ with a finite number of sign changes on [ξ,b]. Second, the function equioscillates n = n(B,r,m,ω,ξ) ≥ r+1 times on the interval [a,b] between -B and B. This analogy with the properties of extremal functions in the linear case ω(t) = t of the problem (*) makes it natural to call these functions the Zolotarev and Chebyshev ω-splines.As in the linear case ω(t)=t, the solution of the problem (*) leads to the qualitative description of extremal functions and sharp additive inequalities for intermediate derivatives in the celebrated Kolmogorov problems on the infinite intervals I = ℝ or ℝ₊:$||f^{(m)}||_{_∞(I)} → sup$, $f ∈ W^rH^ω(I)$, $||f||_{_∞(I)} ≤ B$, 0 < m ≤ r.

CONTENTS0. Introduction...........................................................................................51. Extrema of functionals in $H^ω[a,b]$ and perfect ω-splines................112. Auxiliary results...................................................................................223. Formulation of the main result.............................................................254. Proof of the main result.......................................................................325. The extrapolation problem...................................................................546. Maximization of functionals in $H^ω[a₁,a₂]$, -∞ ≤ a₁ < a₂ ≤ ∞..............567. Euler ω-splines on the finite interval....................................................62Appendix A. Construction of Chebyshev splines.....................................68Appendix B. Construction of Zolotarev splines........................................70References.............................................................................................78

1991 Mathematics Subject Classification: Primary 41A17, 41A44; Secondary 26A15, 26A16, 26A51, 26D10, 46N10, 46N40, 47G10, 52A40, 58C30, 65D07, 65D25, 90C29, 90C26, 90C30.

Détail

Zbl 0928.41009

EUDML-ID : urn:eudml:doc:271243