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

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Rozprawy Matematyczne tom/nr w serii: 379 wydano: 1998
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Warianty tytułu
Abstrakty
EN
Abstract
The 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 $𝓩 = 𝓩_{B,r,m,ω,ξ}$ of the problem (*) enjoys the following two characteristic properties. First, the function $𝓩^{(r)}(·) - 𝓩^{(r)}(ξ)$ 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 𝓩 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 𝓩 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₊:
$||f^{(m)}||_{𝕃_∞(I)} → sup$, $f ∈ W^rH^ω(I)$, $||f||_{𝕃_∞(I)} ≤ B$, 0 < m ≤ r.
EN
CONTENTS
0. Introduction...........................................................................................5
1. Extrema of functionals in $H^ω[a,b]$ and perfect ω-splines................11
2. Auxiliary results...................................................................................22
3. Formulation of the main result.............................................................25
4. Proof of the main result.......................................................................32
5. The extrapolation problem...................................................................54
6. Maximization of functionals in $H^ω[a₁,a₂]$, -∞ ≤ a₁ < a₂ ≤ ∞..............56
7. Euler ω-splines on the finite interval....................................................62
Appendix A. Construction of Chebyshev splines.....................................68
Appendix B. Construction of Zolotarev splines........................................70
References.............................................................................................78
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 379
Liczba stron
81
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXXIX
Daty
wydano
1998
otrzymano
1997-06-02
poprawiono
1998-06-05
Twórcy
Bibliografia
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Uwagi
1991 Mathematics Subject Classification: Primary 41A17, 41A44; Secondary 26A15, 26A16, 26A51, 26D10, 46N10, 46N40, 47G10, 52A40, 58C30, 65D07, 65D25, 90C29, 90C26, 90C30.
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