On the error of approximation by ridge functions with two fixed directions

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We consider the problem of approximation of a continuous multivariate function by sums of two ridge functions in the uniform norm. We obtain a formula for the approximation error in terms functionals generated by closed paths.

[1] V. I. Arnold, On functions of three variables, Dokl. Akad. Nauk SSSR 114 (1957), 679-681; English transl, Amer. Math. Soc. Transl. 28 (1963), 51-54.

[2] M-B. A. Babaev, Sharp estimates for the approximation of functions of several variables by sums of functions of a lesser number of variables, (Russian) Mat. zametki, 12 (1972), 105-114.

[3] D. Braess and A. Pinkus, Interpolation by ridge functions, J.Approx. Theory 73 (1993), 218-236.

[4] R. C. Buck, On approximation theory and functional equations, J.Approx. Theory. 5 (1972), 228-237.

[5] E. J. Candés, Ridgelets: estimating with ridge functions, Ann. Statist. 31 (2003), 1561-1599.

[6] C. K. Chui and X. Li, Approximation by ridge functions and neural networks with one hidden layer, J.Approx. Theory. 70 (1992), 131-141.

[7] W. Dahmen and C. A. Micchelli, Some remarks on ridge functions, Approx. Theory Appl. 3 (1987), 139-143.

[8] S. P. Diliberto and E. G. Straus , On the approximation of a function of several variables by the sum of functions of fewer variables, Pacific J.Math. 1 (1951), 195-210.

[9] D. L. Donoho and I. M. Johnstone, Projection-based approximation and a duality method with kernel methods, Ann. Statist. 17 (1989), 58-106.

[10] J. H .Friedman and W. Stuetzle, Projection pursuit regression, J.Amer. Statist. Assoc. 76 (1981), 817-823.

[11] M. V. Golitschek and W. A. Light , Approximation by solutions of the planar wave equation, Siam J.Numer. Anal. 29 (1992), 816-830.

[12] M. Golomb, Approximation by functions of fewer variables On numerical approximation. Pro- ceedings of a Symposium. Madison 1959. Edited by R.E.Langer. The University of Wisconsin Press. 275-327.

[13] Y. Gordon, V. Maiorov, M. Meyer, S. Reisner, On the best approximation by ridge functions in the uniform norm, Constr. Approx. 18 (2002), 61-85.

[14] S. Ja. Havinson, A Chebyshev theorem for the approximation of a function of two variables by sums of the type ' (x) + (y) ; Izv. Acad. Nauk. SSSR Ser. Mat. 33 (1969), 650-666; English tarnsl. Math. USSR Izv. 3 (1969), 617-632.

[15] P. J. Huber, Projection pursuit, Ann. Statist. 13 (1985), 435-475.

[16] V. E. Ismailov, Approximation by ridge functions and neural networks with a bounded number of neurons, Appl. Anal. 94 (2015), no. 11, 2245-2260.

[17] V. E. Ismailov and A. Pinkus, Interpolation on lines by ridge functions, J. Approx. Theory 175 (2013), 91-113.

[18] V. E. Ismailov, A note on the representation of continuous functions by linear superpositions, Expo. Math. 30 (2012), 96-101.

[19] V. E. Ismailov, On the proximinality of ridge functions, Sarajevo J. Math. 5(17) (2009), no. 1, 109-118.

[20] V. E. Ismailov, On the representation by linear superpositions, J. Approx. Theory 151 (2008), 113-125.

[21] V. E. Ismailov, Characterization of an extremal sum of ridge functions. J. Comput. Appl. Math. 205 (2007), no. 1, 105-115.

[22] V. E. Ismailov, On error formulas for approximation by sums of univariate functions, Int. J. Math. and Math. Sci., volume 2006 (2006), Article ID 65620, 11 pp.

[23] V. E. Ismailov, Methods for computing the least deviation from the sums of functions of one variable, (Russian) Sibirskii Mat. Zhurnal 47 (2006), 1076-1082; translation in Siberian Math. J. 47 (2006), 883-888.

[24] F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Inter- science, New York, 1955.

[25] V. Ya Lin and A.Pinkus, Fundamentality of ridge functions, J.Approx. Theory 75 (1993), 295-311.

[26] B. F. Logan and L.A.Shepp, Optimal reconstruction of a function from its projections, Duke Math.J. 42 (1975), 645-659.

[27] V. Maiorov, R.Meir and J.Ratsaby, On the approximation of functional classes equipped with a uniform measure using ridge functions, J.Approx. Theory 99 (1999), 95-111.

[28] D. E. Marshall and A.G.O'Farrell. Uniform approximation by real functions, Fund. Math. 104 (1979),203-211.

[29] D. E. Marshall and A.G.O'Farrell, Approximation by a sum of two algebras. The lightning bolt principle, J. Funct. Anal. 52 (1983), 353-368.

[30] V. A. Medvedev, Refutation of a theorem of Diliberto and Straus, Mat. zametki, 51(1992), 78-80; English transl. Math. Notes 51(1992), 380-381.

[31] F. Natterer, The Mathematics of Computerized Tomography, Wiley, New York, 1986.

[32] B. Pelletier, Approximation by ridge function fields over compact sets, J.Approx. Theory 129 (2004), 230-239.

[33] P. P. Petrushev, Approximation by ridge functions and neural networks, SIAM J.Math. Anal. 30 (1998), 155-189.

[34] A. Pinkus, Approximating by ridge functions, in: Surface Fitting and Multiresolution Methods, (A.Le Méhauté, C.Rabut and L.L.Schumaker, eds), Vanderbilt Univ.Press (Nashville),1997,279-292.

[35] A. Pinkus, Approximation theory of the MLP model in neural networks, Acta Numerica. 8 (1999), 143-195.

[36] T. J. Rivlin and R. J. Sibner, The degree of approximation of certain functions of two variables by a sum of functions of one variable, Amer. Math. Monthly 72 (1965), 1101-1103.

[37] X. Sun and E. W. Cheney, The fundamentality of sets of ridge functions, Aequationes Math. 44 (1992), 226-235.