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  • Print publication year: 2015
  • Online publication date: August 2015

References

Aczél, J. [1966]: Functional Equations and their Applications, Academic Press, NewYork.
Adams, R. A. [1975]: Sobolev Spaces, Academic Press, New York.
Aumann, G. [1959]: Über approximative Nomographie. II, Bayer. Akad. Wiss. Math.-Nat. Kl. S.-B. 1959, 27–34.
Aumann, G. [1963]: Approximation by step functions, Proc. Amer. Math. Soc. 14, 477–482.
Bauschke, H. H. [1996]: The approximation of fixed points of compositions of nonexpan-sive mappings in Hilbert space, J. Math. Anal. Appl. 202, 150–159.
Bauschke, H. H., Borwein, J. M. [1996]: On projection algorithms for solving convexfeasibility problems, SIAM Review 38, 367–426.
Białynicki-Birula, A., Schinzel, A. [2008]: Representation of multivariate polynomials bysums of univariate polynomials in linear forms, Colloq. Math. 112, 201–233.
Biermann, O. [1903]: Über Näherungsweise Cubaturen, Monat. Math. Phys. 14, 211–225.
Boij, M., Carlini, E., Geramita, A. V. [2011]: Monomials as sums of powers: the realbinary case, Proc. Amer. Math. Soc. 139, 3039–3043.
Boman, J. [1984]: On the closure of spaces of sums of ridge functions and the range of theX-ray transform, Ann. Inst. Fourier (Grenoble) 34, 207–239.
de Boor, C. [2005]: Divided differences, Surveys in Approximation Theory 1, 46–69.[Online article at] http://www.math.technion.ac.il/sat
Brachat, J., Comon, P., Mourrain, B., Tsigaridas, E. [2010]: Symmetric tensor decomposition, Lin. Alg. Appl. 433, 1851–1872.
Braess, D., Pinkus, A. [1993]: Interpolation by ridge functions, J. Approx. Theory 73,218–236.
Browder, F. E. [1967]: Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Zeitschr. 100, 201–225.
Bruck, R. E., Reich, S. [1977]: Nonexpansive projections and resolvents of accretive op-erators in Banach spaces, Houston J. Math. 3, 459–470.
de Bruijn, N. G. [1951]: Functions whose differences belong to a given class, Nieuw Arc.Wisk. 23, 194–218.
de Bruijn, N. G. [1952]: A difference property for Riemann integrable functions and for some similar classes of functions, Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indaga-tiones Math. 14, 145–151.
Buck, R. C. [1972]: On approximation theory and functional equations, J. Approx. Theory 5, 228–237.
Buhmann, M. D., Pinkus, A. [1999]: Identifying linear combinations of ridge functions, Adv. Appl. Math. 22, 103–118.
Candès, E. J. [1998]: Ridgelets: Theory and Applications, Ph. D. dissertation, Dept. Statis-tics, Stanford University.
Candès, E. J. [1999]: Harmonic analysis of neural networks, Appl. Comput. Harmonic Anal. 6, 197–218.
Candès, E. J., Donoho, D. L. [1999]: Ridgelets: a key to higher-dimensional intermit-tency·, Philos T. Royal Soc. A 357, 2495–2509.
Cheney, E. W. [1966]: Introduction to Approximation Theory, McGraw-Hill, New York.
Chlebowicz, A., Wołowiec-Musial, M. [2005]: Forms with a unique representation as asum of powers of linear forms, Tatra Mt. Math. Publ. 32, 33–39.
Chung, K. C., Yao, T. H. [1977]: On lattices admitting unique Lagrange interpolations,SIAM J. Numer. Anal. 14, 735–743.
Cohen, A., Daubechies, I., DeVore, R. A., Kerkyacharian, G., Picard, D. [2012]: Capturingridge functions in high dimensions from point queries, Constr. Approx. 35, 225–243.
Comon, P., Golub, G., Lim, L.-H., Mourrain, B. [2008]: Symmetric tensors and symmetrictensor rank, SIAM J. Matrix Anal. Appl. 30, 1254–1279.
Courant, R., Hilbert, D. [1962] Methods of Mathematical Physics, Vol. II, Interscience Publishers, New York.
Dahmen, W., Micchelli, C. A. [1987]: Some remarks on ridge functions, Approx. Theoryand its Appl. 3, 139–143.
Deutsch, F. [1979]: The alternating method of Von Neumann, in Multivariate Approximation Theory, ISNM 51, 83–96, eds. W, Schempp, K, Zeller, Birkhäuser, Basel.
Deutsch, F., Hundal, H. [1997]: The rate of convergence for the method of alternatingprojections, II, J. Math. Anal. Appl. 205, 381–405.
Deutsch, F., Hundal, H. [2010]: Slow convergence of sequences of linear operators II:arbitrary slow convergence, J. Approx. Theory 162, 1717–1738.
Diaconis, P., Shahshahani, M. [1984]: On nonlinear functions of linear combinations,SIAM J. Sci. Stat. Comput. Applications 5, 175–191.
Diliberto, S. P., Straus, E. G. [1951]: On the approximation of a function of several variables by the sum of functions of fewer variables, Pacific J. Math. 1, 195–210.
Donoho, D. L., Johnstone, I. M. [1989]: Projection-based approximation and a duality method with kernel methods, Ann. Statist. 17, 58–106.
Dyn, N., Light, W. A., Cheney, E. W. [1989]: Interpolation by piecewise-linear radial basis functions, J. Approx. Theory 59, 202–223.
Edwards, R. E. [1965]: Functional Analysis, Theory and Applications, Holt, Rinehart & Winston, New York.
Ellison, W. J. [1971]: Waring's problem, Amer. Math. Monthly 78, 10–36.
Erdélyi, A. (Ed.) [1953]: Higher Transcendental Functions, Vol. 2, Bateman Manuscript project, McGraw-Hill, New York.
Falconer, K. J. [1979]: Consistency conditions for a finite set of projections of a function, Math. Proc. Camb. Phil. Soc. 85, 61–68.
Fornasier, M., Schnass, K., Vybı́ral, J. [2012]: Learning functions of few arbitrary linear parameters in high dimensions, Found. Comput. Math. 12, 229–262.
Franchetti, C., Light, W. [1986]: On the von Neumann alternating algorithm in Hilbertspace, J. Math. Anal. Appl. 114, 305–314.
Friedman, J. H., Stuetzle, W. [1981]: Projection pursuit regression, J. Amer. Statist. Assoc. 76, 817–823.
Garkavi, A. L., Medvedev, V. A., Khavinson, S. Ya. [1995]: On existence of a best uniform approximation of a function in two variables by the sums φ(x)+ψ(y), Sibirsk. Mat. Zh. 36, 819–827; English translation in Siberian Math. J. 36, 707–713.
Golomb, M. [1959]: Approximation by functions of fewer variables, in On Numerical Approximation, 275–327, ed. R., Langer, University of Wiscons in Press, Madison.
Halperin, I. [1962]: The product of projection operators, Acta Sci. Math. (Szeged) 23, 96–99.
Hamaker, C., Solmon, D. C. [1978]: The angles between the null spaces of X-rays, J. Math. Anal. and Appl. 62, 1–23.
Hamel, G. [1905]: Eine Basis aller Zahlen und die unstetigen Lösungen der Funktional-gleichung f(x+ y) = f(x) + f(y), Math. Ann. 60, 459–462.
Hardy, G. H., Littlewood, J. E., Pólya, G. [1952]: Inequalities, 2nd edn, Cambridge University Press.
Helgason, S. [1980]: The Radon Transform, Progress in Mathematics 5, Birkhäuser.
Hilbert, D. [1909]: Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl nter Potenzen (Waringsches Problem), Math. Ann. 67, 281–300.
Horn, R. A., Johnson, C. R. [1991]: Topics in Matrix Analysis, Cambridge University Press.
Huber, P. J. [1985]: Projection pursuit, Ann. Statist. 13, 435–475.
Iarrobino, A. [1995]: Inverse system of a symbolic power II. The Waring problem for forms, J. Algebra 174, 1091–1110.
Ismailov, V. E. [2007a]: A note on the best L2 approximation by ridge functions, Appl. Math. E–Notes 7, 71–76.
Ismailov, V. E. [2007b]: Characterization of an extremal sum of ridge functions, J. Comput. Appl. Math. 205, 105–115.
Ismailov, V. E. [2008a]: On the representation by linear superpositions, J. Approx. Theory 151, 113–125.
Ismailov, V. E. [2009]: On the proximinality of ridge functions, Sarajevo J. Math. 5, 109–118.
Ismailov, V. E. [2014]: Approximation by ridge functions and neural networks with abounded number of neurons, to appear in Appl. Anal.
Ismailov, V. E., Pinkus, A. [2013]: Interpolation on lines by ridge functions, J. Approx. Theory 175, 91–113.
John, F. [1955]: Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience Publishers, Inc., New York.
Jones, L. K. [1987]: On a conjecture of Huber concerning the convergence of projection pursuit regression, Ann. Statist. 15, 880–882.
Jones, L. K. [1992]: A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training, Ann. Statist. 20, 608–613.
Kemperman, J. H. B. [1957]: A general functional equation, Trans. Amer. Math. Society, 86, 28–56.
Khavinson, S. Ya. [1997]: Best Approximation by Linear Superpositions (Approximate Nomography), Transl. Math. Monographs, 159, AMS, Providence, RI.
Kroó, A. [1997]: On approximation by ridge functions, Constr. Approx. 13, 447–460.
Kuczma, M. [1968]: Functional Equations in a Single Variable, PWN – Polish Scientific Publishers, Warszawa.
Lang, H. [1984]: On sums of subspaces in topological vector spaces and an application in theoretical tomography, Appl. Anal. 18, 257–265.
Leshno, M., Lin, V. Ya., Pinkus, A., Schocken, S. [1993]: Multilayer feedforward networks with a non-polynomial activation function can approximate any function, Neural Networks 6, 861–867.
Light, W. A., Cheney, E. W. [1985]: Approximation Theory in Tensor Product Spaces,LNM 1169, Springer-Verlag, Berlin.
Light, W. A., Holland, S. M. [1984]: The L1-version of the Diliberto–Straus algorithm inC(T × S), Proc. Edinburgh Math. Soc. 27, 31–45.
Light, W. A., McCabe, J. H., Phillips, G. M., Cheney, E. W. [1982]: The approximation of bivariate functions by sums of univariate ones using the L1-metric, Proc. Edinburgh Math. Soc. 25, 173–181.
Lin, V. Ya., Pinkus, A. [1993]: Fundamentality of ridge functions, J. Approx. Theory 75, 295–311.
Logan, B. F., Shepp, L. A. [1975]: Optimal reconstruction of a function from its projections, Duke Math. J. 42, 645–659.
Maiorov, V. E. [1999]: On best approximation by ridge functions, J. Approx. Theory 99, 68–94.
Maiorov, V. E. [2010a]: Best approximation by ridge functions in Lp-spaces, Ukr. Math. J. 62, 452–466.
Maiorov, V., Meir, R., Ratsaby, J. [1999]: On the approximation of functional classes equipped with a uniform measure using ridge functions, J. Approx. Theory 99, 95–111.
Maiorov, V. E., Oskolkov, K. I., Temlyakov, V. N. [2002]: Gridge approximation and Radon compass, in Approximation Theory, 284–309, ed. B. D., Bojanov, DARBA, Sofia.
Marshall, D. E., O'Farrell, A. G. [1979]: Uniform approximation by real functions, Fund. Math. 104, 203–211.
Medvedev, V. A. [1991]: On the sum of two closed algebras of continuous functions on acompactum, Funk. Anal. i Pril. 27, 33–36; English translation in Func. Anal. Appl. 27,28–30.
Medvedev, V. A. [1992]: Refutation of a theorem of Diliberto and Straus, Mat. Zametki 51, 78–80; English translation in Math. Notes 51, 380–381.
Mordashev, V. M. [1969]: Best approximations of functions of several variables by sums of functions of fewer variables, Mat. Zametki 5, 217–226; English translation in Math.Notes 5, 132–137.
Murata, N. [1996]: An integral representation of functions using three-layered networksand their approximation bounds, Neural Networks 9, 947–956.
Natterer, F. [1986]: The Mathematics of Computerized Tomography, John Wiley & Sons.
von Neumann, J. [1950]: Functional Operators – Vol. II. The Geometry of Orthogonal Spaces, Annals of Math. Studies #22, Princeton University Press, Princeton, NJ. (Thisis a reprint of mimeographed lecture notes first distributed in 1933.)
Oskolkov, K. I. [1997]: Ridge approximation, Fourier–Chebyshev analysis, and optimal quadrature formulas, Tr. Mat. Inst. Steklova 219, Teor. Priblizh. Garmon. Anal., 269–285; English translation in Proc. Steklov Inst. Math. 219, 265–280.
Oskolkov, K. I. [1999a]: Linear and nonlinear methods for ridge approximation, metric theory of functions and related problems in analysis, 165–195, Izd. Nauchno-Issled.Aktuarno-Finans. Tsentra (AFTs), Moscow, (Russian).
Oskolkov, K. I. [2002]: On representations of algebraic polynomials by superpositions of plane waves, Serdica Math. J. 28, 379–390.
Petersen, B. E., Smith, K. T., Solmon D. C. [1979]: Sums of plane waves, and the rangeof the Radon transform, Math. Ann. 243, 153–161.
Petrushev, P. P. [1998]: Approximation by ridge functions and neural networks, SIAMJ. Math. Anal. 30, 155–189.
Pinkus, A. [1999]: Approximation theory of the MLP model in neural networks, Acta Numerica 8, 143–195.
Pinkus, A. [2013]: Smoothness and uniqueness in ridge function representation, Indagationes Mathematicae 24, 725–738.
Pinkus, A. [2015]: The alternating algorithm in a uniformly convex and uniformly smooth Banach space, J. Math. Anal. Appl. 421, 747–753.
Radon, J. [1948]: Zur mechanischen Kubatur, Monatsh. der Math. Physik 52, 286–300.
Reich, S. [1982]: Nonlinear semigroups, accretive operators, and applications, in Nonlinear Phenomena in Mathematical Sciences, 831–838, ed. V., Lakshmikantham, Academic Press, New York.
Reich, S. [1983]: A limit theorem for projections, Linear and Multilinear Alg. 13, 281–290.
Reznick, B. [1992]: Sums of even powers of real linear forms, Memoirs A. M. S. 463.
Rudin, W. [1973]: Functional Analysis, McGraw-Hill Inc., New York.
Schinzel, A. [2002a]: On a decomposition of polynomials in several variables, J. Théor.Nom. Bordeaux 14, 647–666.
Schinzel, A. [2002b]: On a decomposition of polynomials in several variables, II, Colloq. Math. 92, 67–79.
Schwartz, L. [1944]: Sur certaines familles non fondamentales de fonctions continues, Bull. Soc. Math. France 72, 141–145.
Smith, K. T., Solmon, D. C., Wagner, S. I. [1977]: Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83,1227–1270.
Stahl, D., de Boor, C. [2011]: On Radons recipe for choosing correct sites for multivariate polynomial interpolation, J. Approx. Theory 163, 1854–1858.
Stein, E. M., Weiss, G. [1971]: Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton.
Stridsberg, E. [1912]: Sur la démonstration de M. Hilbert du théorème de Waring, Math.Ann. 72, 145–152.
Sun, X. [1993]: Ridge function spaces and their interpolation property, J. Math. Anal. Appl. 179, 28–40.
Svensson, L. [1989]: Functional analytic approach to stability problems in three-dimensional theoretical tomography, J. Math. Anal. Appl. 139, 303–310.
Sylvester, J. J. [1886]: Sur une extension d'un théorème de Clebsch relatif aux courbes duquatrième degré, C. R. Math. Acad. Sci. Paris 102, 1532–1534.
Temlyakov, V. N. [2000]: Weak greedy algorithms, Adv. Comput. Math 12, 213–227.
Temlyakov, V. N. [2011]: Greedy Approximation, Cambridge Monographs on Applied and Computational Math., Vol. 20, Cambridge University Press.
Tyagi, H., Cevher, V. [2014]: Learning non-parametric basis independent models frompoint queries via low-rank methods, Appl. Comput. Harmonic Anal. 37, 389–412.
Usevich, K. [2014]: Decomposing multivariate polynomials with structured low-rank matrix completion, in 21st International Symposium on Mathematical Theory of Networks and Systems, July 7–11, 2014, Groningen, The Netherlands, 1826–1833.
Vostrecov, B. A. [1963]: Conditions for a function of many variables to be representableas a sum of a finite number of plane waves traveling in given directions, Dokl. Akad.Nauk SSSR 153, 16–19; English translation in Soviet Math. Dokl. 4, 1588–1591.
Vostrecov, B. A., Ignat'eva, A. V. [1967]: The existence of best approximation of functions by sums of a finite number of plane waves of given directions in the Lp metric, Dokl.Akad. Nauk SSSR 176, 1225–1228; English translation in Soviet Math. Dokl. 8, 1288–1291.
Vostrecov, B. A., Kreines, M. A. [1961]: Approximation of continuous functions by superpositions of plane waves, Dokl. Akad. Nauk SSSR 140, 1237–1240; English translationin Soviet Math. Dokl. 2, 1326–1329.
Vostrecov, B. A., Kreines, M. A. [1962]: Approximation of a plane wave by superpositionsof plane waves of given directions, Dokl. Akad. Nauk SSSR 144, 1212–1214; Englishtranslation in Soviet Math. Dokl. 3, 875–877.
Weinmann, A. [1994]: The interpolation problem for ridge functions, Numer. Funct.Anal. Optim. 15, 183–186.