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.
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References
R. A. Aliev and V. E. Ismailov, “On a smoothness problem in ridge function representation, ” Adv. Appl. Math. 73, 154–169 (2016).
D. Braess and A. Pinkus, “Interpolation by ridge functions, ” J. Approx. Theory 73, 218–236 (1993).
N. G. de Bruijn, “Functions whose differences belong to a given class, ” Nieuw Arch. Wiskd., Ser. 2, 23, 194–218 (1951).
M. D. Buhmann and A. Pinkus, “Identifying linear combinations of ridge functions, ” Adv. Appl. Math. 22 (1), 103–118 (1999).
A. Yu. Golovko, “Additive and multiplicative anisotropic estimates for integral norms of differentiable functions on irregular domains, ” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 290, 293–303 (2015) [Proc. Steklov Inst. Math. 290, 277–287 (2015)].
S. Ya. Havinson, “A Chebyshev theorem for the approximation of a function of two variables by sums of the type ϕ(x) + ψ(y), ” Izv. Akad. Nauk SSSR, Ser. Mat. 33 (3), 650–666 (1969) [Math. USSR, Izv. 3, 617–632 (1969)].
H. Herrlich, Axiomof Choice (Springer, Berlin, 2006), Lect. Notes Math. 1876.
V. E. Ismailov and A. Pinkus, “Interpolation on lines by ridge functions, ” J. Approx. Theory 175, 91–113 (2013).
S. V. Konyagin and A. A. Kuleshov, “On the continuity of finite sums of ridge functions, ” Mat. Zametki 98 (2), 308–309 (2015) [Math. Notes 98, 336–338 (2015)].
S. V. Konyagin and A. A. Kuleshov, “On some properties of finite sums of ridge functions defined on convex subsets of Rn, ” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 293, 193–200 (2016) [Proc. Steklov Inst. Math. 293, 186–193 (2016)].
V. E. Maiorov, “On best approximation by ridge functions, ” J. Approx. Theory 99, 68–94 (1999).
Yu. P. Ofman, “Best approximation of functions of two variables by functions of the form ϕ(x) + ψ(y), ” Izv. Akad. Nauk SSSR, Ser. Mat. 25 (2), 239–252 (1961) [Am. Math. Soc. Transl., Ser. 2, 44, 12–28 (1965)].
A. Pinkus, RidgeFunctions (Cambridge Univ. Press, Cambridge, 2015), Cambridge Tracts Math. 205.
X. Sun and E. W. Cheney, “The fundamentality of sets of ridge functions, ” Aequationes Math. 44, 226–235 (1992).
A. I. Tyulenev, “Traces of weighted Sobolev spaces with Muckenhoupt weight. The case p = 1, ” Nonlinear Anal., Theory Methods. Appl. 128, 248–272 (2015).
A. A. Vasil’eva, “Entropy numbers of embedding operators for weighted Sobolev spaces, ” Mat. Zametki 98 (6), 937–940 (2015) [Math. Notes 98, 982–985 (2015)].
A. A. Vasil’eva, “Widths of Sobolev weight classes on a domain with cusp, ” Mat. Sb. 206 (10), 37–70 (2015) [Sb. Math. 206, 1375–1409 (2015)].
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Original Russian Text © A.A. Kuleshov, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 294, pp. 99–104.
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Kuleshov, A.A. On some properties of smooth sums of ridge functions. Proc. Steklov Inst. Math. 294, 89–94 (2016). https://doi.org/10.1134/S0081543816060067
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DOI: https://doi.org/10.1134/S0081543816060067