Lp-norm approximation of Holder functions by harmonic functions 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/spbu01.2023.207

Abstract

In this paper we consider the class of H¨older functions in the sense of Lp norm on certain compacts in Rm (m >= 3) and prove theorems on approximation by functions harmonic in neighborhoods of these compacrs. These compacts are a generalization to the higher dimensions of the concept of chord-arc curve in R3. The size of the neighborhood decreases along with an increase in the accuracy of the approximation. Estimates of the approximation rate and the gradient of the approximation functions are made in the same Lp-norm.

Keywords:

constructive description, H¨older classes, approximation, harmonic functions, chord-arc curves

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References

Литература

1. Моторный В. П. Приближение функцийалгебраическими полиномами в метрике Lp. Известия АН СССР. Сер.: Математика 35 (4), 874-899 (1971).

2. Потапов М. К. О структурных характеристиках классов функцийс данным порядком наилучшего приближения. Труды МИАН СССР 134, 260-277 (1975).

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

4. Алексеева Т. А., Широков Н. А. Классы Гёльдера в Lp-норме на chord-arc кривойв R3. Алгебра и анализ 34 (4), 1-21 (2022).

5. Павлов Д. А. Конструктивное описание гёльдеровых классов на некоторых многомерных компактах. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 8 (66), вып. 3, 430-441 (2021). https://doi.org/10.21638/spbu01.2021.305

References

1. Motornyi V. P. Approximation of functions by algebraic polynomials on the Lp metric. Izvestiya AN SSSR. Seriya Matematika 35, 874-899 (1971). (In Russian)

2. Potapov M. K. The structural characteristic of the classes of functions with a given order of best approximation. Trudy MIAN SSSR. 134, 260-277 (1975). (In Russian)

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

4. Alexeeva T. A., Shirokov N. A. H¨older classes in Lp norm on a chord arc curve in R3. Algebra i analiz 34 (4), 1-21 (2022). (In Russian)

5. Pavlov D. A. Constructive description of Holder classes on some multidimensional compact sets. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 8 (66), iss. 3, 430-441 (2021). https://doi.org/10.21638/spbu01.2021.305 (In Russian) [Eng. transl.: Vestnik St Petersburg University. Mathematics 54, iss. 3, 245-253 (2021). https://doi.org/10.1134/S1063454121030055].

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Published

2023-05-10

How to Cite

Pavlov, D. A. (2023). Lp-norm approximation of Holder functions by harmonic functions on some multidimensional compact sets. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 10(2), 259–269. https://doi.org/10.21638/spbu01.2023.207

Issue

Vol. 10 No. 2 (2023)

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.