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Tr. Mat. Inst. Steklova, 2018, Volume 301, Pages 155–181 (Mi tm3913)  

Some problems in the theory of ridge functions

S. V. Konyagina, A. A. Kuleshovb, V. E. Maiorovc

a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
b Laboratory "Multidimensional Approximation and Applications", Lomonosov Moscow State University, Moscow, 119991 Russia
c Technion – Israel Institute of Technology, Haifa 32000 Israel

Abstract: Let $d\ge2$ and $E\subset\mathbb R^d$ be a set. A ridge function on $E$ is a function of the form $\varphi(\mathbf a\cdot\mathbf x)$, where $\mathbf x=(x_1,…,x_d)\in E$, $\mathbf a=(a_1,…,a_d)\in\mathbb R^d\setminus\{\mathbf0\}$, $\mathbf a\cdot\mathbf x=\sum_{j=1}^da_jx_j$, and $\varphi$ is a real-valued function. Ridge functions play an important role both in approximation theory and mathematical physics and in the solution of applied problems. The present paper is of survey character. It addresses the problems of representation and approximation of multidimensional functions by finite sums of ridge functions. Analogs and generalizations of ridge functions are also considered.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations PRAS-18-01
Ministry of Education and Science of the Russian Federation 14.W03.31.0031
This work is supported by the Program of the Presidium of the Russian Academy of Sciences no. 01 “Fundamental Mathematics and Its Applications” under grant PRAS-18-01 (S.V.K.) and by a grant of the Government of the Russian Federation (project no. 14.W03.31.0031, A.A.K.).


DOI: https://doi.org/10.1134/S0371968518020127

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 301, 144–169

Bibliographic databases:

Document Type: Article
UDC: 517.518.2
Received: December 27, 2017

Citation: S. V. Konyagin, A. A. Kuleshov, V. E. Maiorov, “Some problems in the theory of ridge functions”, Complex analysis, mathematical physics, and applications, Collected papers, Tr. Mat. Inst. Steklova, 301, MAIK Nauka/Interperiodica, Moscow, 2018, 155–181; Proc. Steklov Inst. Math., 301 (2018), 144–169

Citation in format AMSBIB
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\paper Some problems in the theory of ridge functions
\inbook Complex analysis, mathematical physics, and applications
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2018
\vol 301
\pages 155--181
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3913}
\crossref{https://doi.org/10.1134/S0371968518020127}
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\jour Proc. Steklov Inst. Math.
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\vol 301
\pages 144--169
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