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Trudy Mat. Inst. Steklova, 2016, Volume 293, Pages 193–200 (Mi tm3713)  

This article is cited in 8 scientific papers (total in 8 papers)

On some properties of finite sums of ridge functions defined on convex subsets of $\mathbb R^n$

S. V. Konyagin, A. A. Kuleshov

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: Necessary conditions are established for the continuity of finite sums of ridge functions defined on convex subsets $E$ of the space $\mathbb R^n$. It is shown that under some constraints imposed on the summed functions $\varphi _i$, in the case when $E$ is open, the continuity of the sum implies the continuity of all $\varphi _i$. In the case when $E$ is a convex body with nonsmooth boundary, a logarithmic estimate is obtained for the growth of the functions $\varphi _i$ in the neighborhoods of the boundary points of their domains of definition. In addition, an example is constructed that demonstrates the accuracy of the estimate obtained.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.1134/S0371968516020138

Full text: PDF file (199 kB)
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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 293, 186–193

Bibliographic databases:

UDC: 517.518.2
Received: September 18, 2015

Citation: S. V. Konyagin, A. A. Kuleshov, “On some properties of finite sums of ridge functions defined on convex subsets of $\mathbb R^n$”, Function spaces, approximation theory, and related problems of mathematical analysis, Collected papers. In commemoration of the 110th anniversary of Academician Sergei Mikhailovich Nikol'skii, Trudy Mat. Inst. Steklova, 293, MAIK Nauka/Interperiodica, Moscow, 2016, 193–200; Proc. Steklov Inst. Math., 293 (2016), 186–193

Citation in format AMSBIB
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\by S.~V.~Konyagin, A.~A.~Kuleshov
\paper On some properties of finite sums of ridge functions defined on convex subsets of~$\mathbb R^n$
\inbook Function spaces, approximation theory, and related problems of mathematical analysis
\bookinfo Collected papers. In commemoration of the 110th anniversary of Academician Sergei Mikhailovich Nikol'skii
\serial Trudy Mat. Inst. Steklova
\yr 2016
\vol 293
\pages 193--200
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Kuleshov, “On some properties of smooth sums of ridge functions”, Proc. Steklov Inst. Math., 294 (2016), 89–94  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. A. A. Kuleshov, “Continuous Sums of Ridge Functions on a Convex Body and the Class VMO”, Math. Notes, 102:6 (2017), 799–805  mathnet  crossref  crossref  mathscinet  isi  elib
    3. V. E. Ismailov, “A note on the equioscillation theorem for best ridge function approximation”, Expo. Math., 35:3 (2017), 343–349  crossref  mathscinet  zmath  isi  scopus
    4. S. V. Konyagin, A. A. Kuleshov, V. E. Maiorov, “Some problems in the theory of ridge functions”, Proc. Steklov Inst. Math., 301 (2018), 144–169  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. R. A. Aliev, A. A. Asgarova, V. E. Ismailov, “A note on continuous sums of ridge functions”, J. Approx. Theory, 237 (2019), 210–221  crossref  mathscinet  zmath  isi
    6. A. A. Kuleshov, “Continuous sums of ridge functions on a convex body with dini condition on moduli of continuity at boundary points”, Anal. Math., 45:2 (2019), 335–345  crossref  isi
    7. R. A. Aliev, A. A. Asgarova, V. E. Ismailov, “On the Holder continuity in ridge function representation”, Proc. Inst. Math. Mech., 45:1 (2019), 31–40  isi
    8. R. A. Aliev, V. E. Ismailov, “A representation problem for smooth sums of ridge functions”, J. Approx. Theory, 257 (2020), 105448  crossref  mathscinet  zmath  isi
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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