Abstract:
The following problem is studied: If a finite sum of ridge functions defined on an open subset of Rn belongs to some smoothness class, can one represent this sum as a sum of ridge functions (with the same set of directions) each of which belongs to the same smoothness class as the whole sum? It is shown that when the sum contains m terms and there are m−1 linearly independent directions among m linearly dependent ones, such a representation exists.
Citation:
A. A. Kuleshov, “On some properties of smooth sums of ridge functions”, Modern problems of mathematics, mechanics, and mathematical physics. II, Collected papers, Trudy Mat. Inst. Steklova, 294, MAIK Nauka/Interperiodica, Moscow, 2016, 99–104; Proc. Steklov Inst. Math., 294 (2016), 89–94
\Bibitem{Kul16}
\by A.~A.~Kuleshov
\paper On some properties of smooth sums of ridge functions
\inbook Modern problems of mathematics, mechanics, and mathematical physics.~II
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2016
\vol 294
\pages 99--104
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\jour Proc. Steklov Inst. Math.
\yr 2016
\vol 294
\pages 89--94
\crossref{https://doi.org/10.1134/S0081543816060067}
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https://www.mathnet.ru/eng/tm3735
https://doi.org/10.1134/S0371968516030067
https://www.mathnet.ru/eng/tm/v294/p99
This publication is cited in the following 7 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, 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
Citation in format AMSBIB
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