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Mat. Zametki, 2001, Volume 70, Issue 5, Pages 758–768 (Mi mz787)  

This article is cited in 5 scientific papers (total in 5 papers)

On a Characterization of Spaces of Differentiable Functions

A. N. Morozov

P. G. Demidov Yaroslavl State University

Abstract: In this paper, we generalize Bernstein's theorem characterizing the space $C^k[a,b]$ by means of local approximations. The closed interval $[a,b]$ is partitioned into disjoint half-intervals on which best approximation polynomials of degree $k-1$ divided by the lengths of these half-intervals taken to the power $k$ are considered. The existence of the limits of these ratios as the lengths of the half-intervals tend to zero is a criterion for the existence of the $k$th derivative of a function. We prove the theorem in a stronger form and extend it to the spaces $W_p^k[a,b]$.

DOI: https://doi.org/10.4213/mzm787

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English version:
Mathematical Notes, 2001, 70:5, 688–697

Bibliographic databases:

UDC: 517.5
Received: 18.11.1996
Revised: 25.01.2000

Citation: A. N. Morozov, “On a Characterization of Spaces of Differentiable Functions”, Mat. Zametki, 70:5 (2001), 758–768; Math. Notes, 70:5 (2001), 688–697

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. N. Morozov, “$K$-funktsionaly i nailuchshie kusochno-polinomialnye priblizheniya”, Model. i analiz inform. sistem, 14:1 (2007), 27–30  mathnet
    2. A. N. Morozov, “Opisanie prostranstv differentsiruemykh funktsii pri pomoschi lokalnykh priblizhenii”, Model. i analiz inform. sistem, 16:1 (2009), 7–15  mathnet
    3. A. N. Morozov, “Local Approximations of Differentiable Functions”, Math. Notes, 100:2 (2016), 256–262  mathnet  crossref  crossref  mathscinet  isi  elib
    4. A. N. Morozov, “O differentsiruemosti po Teiloru v prostranstvakh $L_p, 0<p\leq \infty$”, Model. i analiz inform. sistem, 25:3 (2018), 323–330  mathnet  crossref  elib
    5. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49  mathnet
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