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 Mat. Sb., 2003, Volume 194, Number 1, Pages 103–120 (Mi msb708)

Almost normed spaces of functions with polynomial asymptotic behaviour

L. D. Kudryavtsev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider a linear space of functions asymptotically approaching polynomials of degree not higher than a fixed one as the independent variable approaches infinity. Such a space cannot be normed if the functions in it possess a certain smoothness. For this reason the concept of almost normed space is introduced and the spaces in question, namely, spaces of functions asymptotically or strongly asymptotically approaching polynomials, are shown to be almost normed. The completeness of these spaces in the metric generated by their almost norm is also proved, the connection between the asymptotic approach and the strong asymptotic approach of functions to polynomials is studied, and a new (and shorter) proof of the criterion for the asymptotic approach of functions to polynomials is presented.

DOI: https://doi.org/10.4213/sm708

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English version:
Sbornik: Mathematics, 2003, 194:1, 105–122

Bibliographic databases:

Document Type: Article
UDC: 517.911
MSC: 46E10, 46E40

Citation: L. D. Kudryavtsev, “Almost normed spaces of functions with polynomial asymptotic behaviour”, Mat. Sb., 194:1 (2003), 103–120; Sb. Math., 194:1 (2003), 105–122

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb708
• https://doi.org/10.4213/sm708
• http://mi.mathnet.ru/eng/msb/v194/i1/p103

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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. L. D. Kudryavtsev, “Operators for almost normed spaces”, Dokl. Math., 71:2 (2005), 228–230
2. L. D. Kudryavtsev, “Almost Normed and Quasinormed Spaces”, Proc. Steklov Inst. Math., 248 (2005), 125–137
3. A. A. Bolibrukh, V. A. Il'in, S. M. Nikol'skii, V. M. Filippov, G. N. Yakovlev, “Lev Dmitrievich Kudryavtsev (on his 80th birthday)”, Russian Math. Surveys, 60:1 (2005), 177–188
4. L. D. Kudryavtsev, “Lagrangian asymptotic behaviour of solutions of inhomogeneous systems of ordinary differential equations”, Sb. Math., 197:9 (2006), 1341–1351
5. L. D. Kudryavtsev, “Problems with asymptotic initial data for systems of ordinary differential equations”, Dokl. Math., 73:2 (2006), 202
6. Granata A., “Polynomial asymptotic expansions in the real domain: the geometric, the factorizational, and the stabilization approaches”, Anal. Math., 33:3 (2007), 161–198
7. S. A. Belyaev, “Solutions of ordinary differential equations asymptotically approaching polynomials in the reciprocal of an independent variable”, Dokl. Math., 80:1 (2009), 581
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