## Tbilisi Mathematical Journal

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

#### Abstract

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.

#### Article information

Source
Tbilisi Math. J., Volume 10, Issue 2 (2017), 111-120.

Dates
Accepted: 15 March 2017
First available in Project Euclid: 26 May 2018

https://projecteuclid.org/euclid.tbilisi/1527300048

Digital Object Identifier
doi:10.1515/tmj-2017-0030

Mathematical Reviews number (MathSciNet)
MR3652302

Zentralblatt MATH identifier
1364.41007

#### Citation

Asgarova, Aida Kh.; Babayev, Arzu M-B.; Maharov, Ibrahim K. On the error of approximation by ridge functions with two fixed directions. Tbilisi Math. J. 10 (2017), no. 2, 111--120. doi:10.1515/tmj-2017-0030. https://projecteuclid.org/euclid.tbilisi/1527300048

#### References

• 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.
• 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.
• D. Braess and A. Pinkus, Interpolation by ridge functions, J.Approx. Theory 73 (1993), 218-236.
• R. C. Buck, On approximation theory and functional equations, J.Approx. Theory. 5 (1972), 228-237.
• E. J. Candés, Ridgelets: estimating with ridge functions, Ann. Statist. 31 (2003), 1561-1599.
• C. K. Chui and X. Li, Approximation by ridge functions and neural networks with one hidden layer, J.Approx. Theory. 70 (1992), 131-141.
• W. Dahmen and C. A. Micchelli, Some remarks on ridge functions, Approx. Theory Appl. 3 (1987), 139-143.
• 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.
• D. L. Donoho and I. M. Johnstone, Projection-based approximation and a duality method with kernel methods, Ann. Statist. 17 (1989), 58-106.
• J. H.Friedman and W. Stuetzle, Projection pursuit regression, J.Amer. Statist. Assoc. 76 (1981), 817-823.
• M. V. Golitschek and W. A. Light, Approximation by solutions of the planar wave equation, Siam J.Numer. Anal. 29 (1992), 816-830.
• M. Golomb, Approximation by functions of fewer variables On numerical approximation. Proceedings of a Symposium. Madison 1959. Edited by R.E.Langer. The University of Wisconsin Press. 275-327.
• 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.
• S. Ja. Havinson, A Chebyshev theorem for the approximation of a function of two variables by sums of the type $\varphi \left( {x}\right) +\psi \left( {y}\right) ,$ Izv. Acad. Nauk. SSSR Ser. Mat. 33 (1969), 650-666; English tarnsl. Math. USSR Izv. 3 (1969), 617-632.
• P. J. Huber, Projection pursuit, Ann. Statist. 13 (1985), 435-475.
• V. E. Ismailov, Approximation by ridge functions and neural networks with a bounded number of neurons, Appl. Anal. 94 (2015), no. 11, 2245-2260.
• V. E. Ismailov and A. Pinkus, Interpolation on lines by ridge functions, J. Approx. Theory 175 (2013), 91-113.
• V. E. Ismailov, A note on the representation of continuous functions by linear superpositions, Expo. Math. 30 (2012), 96-101.
• V. E. Ismailov, On the proximinality of ridge functions, Sarajevo J. Math. 5(17) (2009), no. 1, 109-118.
• V. E. Ismailov, On the representation by linear superpositions, J. Approx. Theory 151 (2008), 113-125.
• V. E. Ismailov, Characterization of an extremal sum of ridge functions. J. Comput. Appl. Math. 205 (2007), no. 1, 105–115.
• 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.
• 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.
• F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York, 1955.
• V. Ya Lin and A.Pinkus, Fundamentality of ridge functions, J.Approx. Theory 75 (1993), 295-311.
• B. F. Logan and L.A.Shepp, Optimal reconstruction of a function from its projections, Duke Math.J. 42 (1975), 645-659.
• 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.
• D. E. Marshall and A.G.O'Farrell. Uniform approximation by real functions, Fund. Math. 104 (1979),203-211.
• 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.
• 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.
• F. Natterer, The Mathematics of Computerized Tomography, Wiley, New York, 1986.
• B. Pelletier, Approximation by ridge function fields over compact sets, J.Approx. Theory 129 (2004), 230-239.
• P. P. Petrushev, Approximation by ridge functions and neural networks, SIAM J.Math. Anal. 30 (1998), 155-189.
• 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.
• A. Pinkus, Approximation theory of the MLP model in neural networks, Acta Nume-rica. 8 (1999), 143-195.
• 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.
• X. Sun and E. W. Cheney, The fundamentality of sets of ridge functions, Aequationes Math. 44 (1992), 226-235.