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Quasi-equivalence of bases in some Whitney spaces

Part of: Approximations and expansions Linear function spaces and their duals Topological linear spaces and related structures

Published online by Cambridge University Press:  18 May 2021

Alexander Goncharov  and
Yasemin Şengül
Alexander Goncharov*
Affiliation:
Department of Mathematics, Bilkent University, 06800Ankara, Turkey
Yasemin Şengül
Affiliation:
Faculty of Engineering and Natural Sciences, Sabanci University, Orta Mahalle, Tuzla, 34956 Istanbul, Turkey e-mail: yaseminsengul@sabanciuniv.edu
*
e-mail: goncha@fen.bilkent.edu.tr
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Abstract

If the logarithmic dimension of a Cantor-type set K is smaller than $1$ , then the Whitney space $\mathcal {E}(K)$ possesses an interpolating Faber basis. For any generalized Cantor-type set K, a basis in $\mathcal {E}(K)$ can be presented by means of functions that are polynomials locally. This gives a plenty of bases in each space $\mathcal {E}(K)$ . We show that these bases are quasi-equivalent.

Keywords

Topological basesWhitney spacesquasi-equivalence

MSC classification

Primary: 46A35: Summability and bases
Secondary: 46E10: Topological linear spaces of continuous, differentiable or analytic functions 41A10: Approximation by polynomials
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 106 - 115
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The research was partially supported by TÜBİTAK (Scientific and Technological Research Council of Turkey), Project 119F023.

References

Alpseymen, M., A generalization of Dragilev’s theorem. J. Reine Angew. Math. 276(1975), 124–129.Google Scholar
Aytuna, A., Djakov, P. B., Goncharov, A. P., Terzioğlu, T., and Zahariuta, V. P., Some open problems in the theory of locally convex spaces. Linear Topol. Spaces Complex Anal. 1(1994), 147–165.Google Scholar
Chalov, P. A. and Zahariuta, V. P., On quasidiagonal isomorphisms of generalized power spaces. Linear Topol. Spaces Complex Anal. 2(1995), 35–44.Google Scholar
Crone, L. and Robinson, W. B., Every nuclear Fréchet space with a regular basis has the quasi-equivalence property. Stud. Math. 52(1974/75), 203–207.CrossRefGoogle Scholar
Dragilev, M. M., Standard form of basis for the space of analytic functions (in Russian). Uspekhi Mat. Nauk 15(1960), no. 2 (92), 181–188.Google Scholar
Goncharov, A., Bases in the spaces of ${C}^{\infty }$ -functions on Cantor-type sets. Constr. Approx. 23(2006), 351–360.10.1007/s00365-005-0598-5CrossRefGoogle Scholar
Goncharov, A. and Şengül, Y., Logarithmic dimension and bases in Whitney spaces, submitted.Google Scholar
Goncharov, A., Terzioğlu, T., and Zahariuta, V., On isomorphic classification of tensor products ${E}_{\infty }(a)\ \hat{\otimes}\ {E}_{\infty}^{\prime }(b)$ . Dissertationes Math. (Rozprawy Mat.) 350(1996), 27 pp.Google Scholar
Goncharov, A. and Ural, Z., Mityagin extension problem. Progress report. J. Math. Anal. Appl. 448(2017), no. 1, 357–375.10.1016/j.jmaa.2016.11.001CrossRefGoogle Scholar
Kondakov, V. P., On quasi-equivalence of regular bases in Köthe spaces (in Russisan). Matem. Analiz i ego Pril. 5(1974), 210–213, Rostov-on-Don.Google Scholar
Lindenstrauss, J. and Zippin, M., Banach spaces with a unique unconditional basis. J. Funct. Anal. 3(1969), 115–125.CrossRefGoogle Scholar
Mitjagin, B. S., Fréchet spaces with a unique unconditional basis. Stud. Math. 38(1970), 23–34.CrossRefGoogle Scholar
Mitjagin, B. S., Equivalence of bases in Hilbert scales (in Russian). Stud. Math. 37(1970/71), 111–137.Google Scholar
Mityagin, B. S., Approximate dimension and bases in nuclear spaces. Russ. Math. Surv. 16(1961), no. 4, 59–127.10.1070/RM1961v016n04ABEH004109CrossRefGoogle Scholar
Zaharjuta, V. P., The isomorphism and quasiequivalence of bases for exponential Köthe spaces (in Russian). Dokl. Akad. Nauk SSSR 221(1975), no. 4, 772–774.Google Scholar
Zobin, N., Some remarks on quasi-equivalence of bases in Fréchet spaces. Linear Algebra Appl. 307(2000), no. 1–3, 47–67.10.1016/S0024-3795(99)00277-3CrossRefGoogle Scholar