Abstract:
Necessary conditions are established for the continuity of finite sums of ridge functions defined on convex subsets E of the space Rn. It is shown that under some constraints imposed on the summed functions φi, in the case when E is open, the continuity of the sum implies the continuity of all φi. In the case when E is a convex body with nonsmooth boundary, a logarithmic estimate is obtained for the growth of the functions φi 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.
Citation:
S. V. Konyagin, A. A. Kuleshov, “On some properties of finite sums of ridge functions defined on convex subsets of Rn”, 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
\Bibitem{KonKul16}
\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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\crossref{https://doi.org/10.1134/S0371968516020138}
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\jour Proc. Steklov Inst. Math.
\yr 2016
\vol 293
\pages 186--193
\crossref{https://doi.org/10.1134/S0081543816040131}
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Linking options:
https://www.mathnet.ru/eng/tm3713
https://doi.org/10.1134/S0371968516020138
https://www.mathnet.ru/eng/tm/v293/p193
This publication is cited in the following 11 articles:
Rashid A. Aliev, Fidan M. Isgandarli, “On the representability of a continuous multivariate function by sums of ridge functions”, Journal of Approximation Theory, 304 (2024), 106105
R. A. Aliev, A. A. Asgarova, V. E. Ismailov, “On the representation by bivariate ridge functions”, Ukr. Math. J., 73:5 (2021), 675–685
R. A. Aliev, A. A. Asgarova, V. E. Ismailov, “On the representation by bivariate ridge functions”, Ukr. Mat. Zhurn., 73:5 (2021), 579
R. A. Aliev, V. E. Ismailov, “A representation problem for smooth sums of ridge functions”, J. Approx. Theory, 257 (2020), 105448
R. A. Aliev, A. A. Asgarova, V. E. Ismailov, “A note on continuous sums of ridge functions”, J. Approx. Theory, 237 (2019), 210–221
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
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
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
A. A. Kuleshov, “Continuous Sums of Ridge Functions on a Convex Body and the Class VMO”, Math. Notes, 102:6 (2017), 799–805
V. E. Ismailov, “A note on the equioscillation theorem for best ridge function approximation”, Expo. Math., 35:3 (2017), 343–349
A. A. Kuleshov, “On some properties of smooth sums of ridge functions”, Proc. Steklov Inst. Math., 294 (2016), 89–94
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