Constructive description of Hölder classes on some multidimensional compact sets

Authors

  • Dmitriy A. Pavlov Herzen State Pedagogical University of Russia, 48, nab. r. Moiki, St. Petersburg, 191186, Russian Federation

DOI:

https://doi.org/10.21638/spbu01.2021.305

Abstract

We give a constructive description of Holder classes of functions on certain compacts in Rm (m > 3) in terms of a rate of approximation by harmonic functions in shrinking neighborhoods of these compacts. The considered compacts are a generalization to the higher dimensions of compacts that are subsets of a chord-arc curve in R3. The size of the neighborhood is directly related to the rate of approximation - it shrinks when the approximation becomes more accurate. In addition to being harmonic in the neighborhood of the compact the approximation functions have a property that looks similar to H¨older condition. It consists in the fact that the difference in values at two points is estimated in terms of the size of the neighborhood, if the distance between these points is commensurate with the size of the neighborhood (and therefore it is estimated in terms of the distance between the points).

Keywords:

constructive description, Holder classes, approximation, harmonic functions, chord-arc curves

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References

Литература

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5. Alexeeva T.A., Shirokov N.A. Constructive description of Holder-like classes on an arc in R3 by means of harmonic functions. Journal of Approximation Theory 249, 105308 (2020). https://doi.org/10.1016/j.jat.2019.105308

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7. Dyn’kin E.M. The Pseudoanalytic Extension. Journal d’Analyse Mathematique 60, 45–70 (1993). https://doi.org/10.1007/BF03341966

8. Михлин С.Г. Курс математической физики. Москва, Наука (1968).

References

1. Dzyadyk V.K. Constructive characterization of functions satisfying the condition Lip a(0 < a < 1) on a finite segment of the real axis. Izv. Akad. Nauk SSSR, ser. mat. 20 (5), 623–642 (1956). (In Russian)

2. Andrievskii V.V. The geometric structure of regions, and direct theorems of the constructive theory of functions. Mat. Sb. 126 (168) (1), 41–58 (1985). (In Russian) [Engl. transl.: Math. USSR-Sb. 54 (1), 39–56 (1986). http://dx.doi.org/10.1070/SM1986v054n01ABEH002959].

3. Shirokov N.A. Approximation entropy of continua. Dokl. Akad. Nauk SSSR 235 (3), 546–549 (1977). (In Russian)

4. Shirokov N.A. Constructive descriptions of functional classes by polynomial approximations. Journal of Mathematical Sciences 105, 2269–2291 (2001). http://dx.doi.org/10.1023/A:1011393428151

5. Alexeeva T.A., Shirokov N.A. Constructive description of Holder-like classes on an arc in R3 by means of harmonic functions. Journal of Approximation Theory 249, 105308 (2020). https://doi.org/10.1016/j.jat.2019.105308

6. Pavlov D.A. Constructive description of H¨older classes on compact subsets of R3. Zapiski Nauchnykh Seminarov POMI 491, 119–144 (2020). (In Russian)

7. Dyn’kin E.M. The Pseudoanalytic Extension. Journal d’Analyse Mathematique 60, 45–70 (1993). https://doi.org/10.1007/BF03341966

8. Mikhlin S.G. Mathematical physics course. Moscow, Nauka Publ. (1968). (In Russian)

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Published

2021-09-26

How to Cite

Pavlov, D. A. (2021). Constructive description of Hölder classes on some multidimensional compact sets. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(3), 430–441. https://doi.org/10.21638/spbu01.2021.305

Issue

Vol. 8 No. 3 (2021)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.