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On constrained Markov-Nikolskii and Bernstein type inequalitiesKlurman, Oleksiy 01 September 2011 (has links)
This thesis is devoted to polynomial inequalities with constraints. We present a history of the development of this subject together with recent progress.
In the first part, we solve an analog of classical Markov's problem for monotone polynomials. More precisely, if ∆n denotes the set of all monotone polynomials on [-1,1] of degree n, then for Pn ϵ ∆n and x ϵ [-1,1] the following sharp inequalities hold:
│P’n(x)│≤ 2 max(Sk(-x),Sk(-x))║Pn║,
for n = 2k + 2, k ≥ 0, and
│P'n(x)│ ≤ 2 max (Fk(x), Hk(x))║Pn║,
for n = 2k + 1, k ≥ 0, where
Sk(x) := (1+x)∑_(l=0 )^k▒(J_l (0,1)(x^2)) ;
S_k (x) &:=(1+x)\sum\limits_{l=0}^{k} (J^{(0,1)}_l (x))^2;\\
H_k (x) &:=(1-x^2)\sum\limits_{l=0}^{k-1} (J_l ^{(1,1)} (x))^2;\\
F_k(x) &:=\sum\limits_{l=0}^{k} (J_l ^{(0,0)} (x))^2,
\end{align*}
and $J_l^{(\alpha,\beta)}(x),$ $l\ge 1$ are the Jacobi polynomials.
Let ∆n(1) be the set of all monotone nonnegative polynomials on $[-1,1]$ of degree $n.$ In the second part, we investigate the asymptotic behavior of the constants $$M_{q,p}^{(1)}(n,1):=\sup_{P_n\in\triangle^{(1)}_n}\frac{\|P'_n\|_{L_q
[-1,1]}}{\|P_n\|_{L_p [-1,1]}},$$ in constrained Markov-Nikolskii type inequalities.
Our conjecture is that
\[M^{(1)}_{q,p} (n,1)\asymp
\left\{
\begin{array}{ll}
n^{2+2/p-2/q} , & \mbox{\rm if } 1>1/q-1/p ,\\
\log{n} , & \mbox{\rm if } 1=1/q-1/p, \\
1 , & \mbox{\rm if } 1< 1/q - 1/p .
\end{array}
\right.
\]
We prove this conjecture for all values of p,q > 0, except for the case 0 < q < 1, 1/2 ≤ 1/q- 1/p ≤ 1, p ≠ 1
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Approximation to summable functionsHowlett, Philip George January 1970 (has links)
vi, 150 leaves / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematics, 1971
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Optimal approximations of functions one sided approximation and extrema preserving approximations /Kammerer, W. J. January 1959 (has links)
Thesis (Ph. D.)--University of Wisconsin--Madison, 1959. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 62-68).
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Das Näherungsverfahren Xn̳=[phi](Xn̳-̳1̳) und seine Anwendung auf Theorie und Praxis algebraischer und transzendenter GleichungenVermeil, Hermann, January 1914 (has links)
Thesis (doctoral)--Universität Leipzig, 1914. / On t.p. "n̳" and "n̳-̳1̳" are subscript. Vita. Includes bibliographical references.
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Beste einseitige L-Approximation mit Quasi-Blending-FunktionenKlinkhammer, John. January 2002 (has links) (PDF)
Duisburg, Univ., Diss., 2001.
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Approximate computationBakst, Aaron, January 1937 (has links)
Thesis (Ph. D.)--Columbia University, 1937. / Vita. Bibliography: p. 284-287.
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An investigation of Huynh's normal approximation procedure for estimating criterion-referenced reliabilityPeng, J. Chao-ying. January 1900 (has links)
Thesis--University of Wisconsin--Madison. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 84-88).
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Ein Triangulierungsverfahren zur Approximation mit Dahmen-Micchelli-Seidel-SplinesHussmann, Markus. January 1999 (has links)
Duisburg, Universiẗat, Diss., 1999. / Dateiformat: zip, Dateien in unterschiedlichen Formaten.
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Vertical and Orthogonal L1 Linear Approximation: Analysis and AlgorithmsYamamoto, Peter J. January 1988 (has links)
Note:
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Kantorovich's general theory of approximation methodsWolkowicz, Henry. January 1975 (has links)
No description available.
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