Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-22T20:11:31.243Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  06 July 2018

V. Temlyakov
Affiliation:
University of South Carolina
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akhiezer, N.I. (1965). Lectures in Approximation Theory. Nauka, 1965; English translation of 1st edition published by Ungar, 1956.Google Scholar
Akhiezer, N.I. and Krein, M.G. (1937). On the best approximation of differentiable periodic functions by trigonometric sums. Dokl. Akad. Nauk SSSR, 15 107112.Google Scholar
Andrianov, A.V. and Temlyakov, V.N. (1997). On two methods of generalization of properties of univariate function systems to their tensor product. Trudy MIAN, 219 32–43; English translation in Proc. Steklov Inst. Math., 219 2535.Google Scholar
Babadzhanov, S.B. and Tikhomirov, V.M. (1967). On widths of a certain class in the Lp-spaces (p ≥ 1). Izv. Akad. Nauk UzSSR Ser. Fiz.– Mat. Nauk, 11 2430.Google Scholar
Babenko, K.I. (1985). Some problems in approximation theory and numerical analysis. Russian Math. Surveys, 40 130.CrossRefGoogle Scholar
Babenko, K.I. (1960a). On the approximation of periodic functions of several variables by trigonometric polynomials. Dokl. Akad. Nauk SSSR, 132 247250; English translation in Soviet Math. Dokl., 1 (1960).Google Scholar
Babenko, K.I. (1960b). On the approximation of a certain class of periodic functions of several variables by trigonometric polynomials. Dokl. Akad. Nauk SSSR, 132 982985; English translation in Soviet Math. Dokl., 1 (1960).Google Scholar
Baishanski, B.M. (1983). Approximation by polynomials of given length. Illinois J. Math., 27 449458.CrossRefGoogle Scholar
Bakhvalov, N.S. (1959). On the approximate computation of multiple integrals. Vestnik Moskov. Univ. Ser. Mat. Mekh. Astr. Fiz. Khim., 4 318.Google Scholar
Bakhvalov, N.S. (1963a). Embedding theorems for classes of functions with several bounded derivatives. Vestnik Moskov. Univ. Ser. Mat. Mekh., 3 716.Google Scholar
Bakhvalov, N.S. (1963b). Optimal convergence bounds for quadrature processes and integration methods of Monte Carlo type for classes of functions. Zh. Vychisl. Mat. i Mat. Fiz. Suppl., 4 563.Google Scholar
Bakhvalov, N.S. (1972). Lower estimates of asymptotic characteristics of classes of functions with dominating mixed derivative. Matem. Zametki, 12 655664; English translation in Math. Notes, 12 (1972).Google Scholar
Bary, N.K. (1961). Trigonometric Series. Nauka. English translation, Pergamon Press, 1964.Google Scholar
Bass, R.F. (1988). Probability estimates for multiparameter Brownian processes. Ann. Probab., 16 251264.CrossRefGoogle Scholar
Beck, J. and Chen, W. (1987). Irregularities of Distribution. Cambridge University Press.CrossRefGoogle Scholar
Belinskii, E.S. (1987). Approximation by a “floating” system of exponentials on classes of smooth periodic functions. Matem. Sb., 132 2027; English translation in Math. USSR Sb., 60 (1988).Google Scholar
Belinskii, E.S. (1988). Approximation by a “floating” system of exponentials on classes of periodic functions with bounded mixed derivative. In Research on the Theory of Functions of Many Real Variables. Yaroslavl’ State University, 1633 (in Russian).Google Scholar
Belinskii, E.S. (1989). Approximation of functions of several variables by trigonometric polynomials with given number of harmonics, and estimates of ε-entropy. Anal. Math., 15 6774.CrossRefGoogle Scholar
Belinskii, E.S. (1998a). Decomposition theorems and approximation by a “floating” system of exponentials. Trans. Amer. Math. Soc., 350 4353.CrossRefGoogle Scholar
Belinskii, E.S. (1998b). Estimates of entropy numbers and Gaussian measures for classes of functions with bounded mixed derivative. J. Approx. Theory, 93 114127.CrossRefGoogle Scholar
Bernstein, S.N. (1912). Sur la valeur asymptotique de la meilleure approximation de |x|. Comptes Rendus, 154 184186.Google Scholar
Bernstein, S.N. (1914). Sur la meilleure approximation de |x| par des polynomes des degrés donnés. Acta Math., 37 157.CrossRefGoogle Scholar
Bernstein, S.N. (1952). Collected Works, Vols. I and II. Akad. Nauk SSSR.Google Scholar
Bilyk, D. and Lacey, M. (2008). On the small ball inequality in three dimensions. Duke Math J., 143 81115.CrossRefGoogle Scholar
Bilyk, D., Lacey, M., and Vagharshakyan, A. (2008). On the small ball inequality in all dimensions. J. Funct. Analysis, 254 24702502.CrossRefGoogle Scholar
Binev, P., Cohen, A., Dahmen, W., DeVore, R., and Temlyakov, V.N. (2005). Universal algorithms for learning theory. Part I: Piecewise constant functions. J. Machine Learning Theory, 6 12971321.Google Scholar
Bourgain, J. and Milman, V.D. (1987). New volume ratio properties for convex symmetric bodies in Rn. Invent. Math., 88 319340.CrossRefGoogle Scholar
Bugrov, Ya. S. (1964). Approximation of a class of functions with a dominating mixed derivative. Mat. Sb., 64 410418.Google Scholar
Bungartz, H.-J. and Griebel, M. (1999). A note on the complexity of solving Poisson’s equation for spaces of bounded mixed derivatives. J. Complexity, 15 167199.CrossRefGoogle Scholar
Bungartz, H.-J. and Griebel, M. (2004). Sparse grids. Acta Numerica, 13 147269.CrossRefGoogle Scholar
Bykovskii, V.A. (1985). On the correct order of the error of optimal cubature formulas in spaces with dominating derivative, and on quadratic deviations of grids. Preprint, Computing Center, Far-Eastern Scientific Center, Acad. Sci. USSR, Vladivostok.Google Scholar
Bykovskii, V.A. (1995). Estimates of deviations of optimal lattices in the Lp-norm and the theory of quadrature formulas. Preprint, Applied Mathematics Institute, Far-Eastern Scientific Center, Acad. Sci. Russia, Khabarovsk.Google Scholar
Cassels, J.W.S. (1971). An Introduction to the Geometry of Numbers. Springer-Verlag.Google Scholar
Chazelle, B. (2000). The Discrepancy Method. Cambridge University Press.CrossRefGoogle Scholar
Chebyshev, P.L. (1854). Théorie des mecanismes connus sous le nom de parallélogrammes. Mem. Présentés à l’Acad. Imp. Sci. St.-Pétersbourg par Divers Savants, 7 539568.Google Scholar
Chen, W.W.L. (1980). On irregularities of distribution. Mathematika, 27 153170.CrossRefGoogle Scholar
Cohen, A., DeVore, R.A., and Hochmuth, R. (2000). Restricted nonlinear approximation. Constructive Approx., 16 85113.CrossRefGoogle Scholar
Dai, W. and Milenkovic, O. (2009). Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inf. Theory, 55 22302249.CrossRefGoogle Scholar
Davenport, H. (1956). Note on irregularities of distribution. Mathematika, 3 131135.CrossRefGoogle Scholar
Davis, G., Mallat, S., and Avellaneda, M. (1997). Adaptive greedy approximations. Constructive Approx., 13 5798.CrossRefGoogle Scholar
de la Vallée Poussin, Ch. (1908). Sur la convergence des formules d’interpolation entre ordonées equidistantes. Bull. Acad. Belgique 4 403410.Google Scholar
de la Vallée Poussin, Ch. (1919). Lecons sur l’Approximation des Fonctions d’une Variable Réelle. Gauthier-Villars, Paris, 1919; 2nd edition published by Chelsea Publishing Co., 1970.Google Scholar
DeVore, R.A. (1998). Nonlinear approximation. Acta Numerica 7 51150.CrossRefGoogle Scholar
DeVore, R.A and Lorentz, G.G. (1993). Constructive Approximation. Springer-Verlag.CrossRefGoogle Scholar
DeVore, R.A. and Temlyakov, V.N. (1995). Nonlinear approximation by trigonometric sums. J. Fourier Anal. Applic., 2 2948.CrossRefGoogle Scholar
Dilworth, S.J., Kutzarova, D., and Temlyakov, V.N. (2002). Convergence of some greedy algorithms in Banach spaces. J. Fourier Anal. Applic. 8 489505.CrossRefGoogle Scholar
Dilworth, S.J., Kalton, N.J., and Kutzarova, Denka (2003a). On the existence of almost greedy bases in Banach spaces. Studia Math., 158 67101.CrossRefGoogle Scholar
Dilworth, S.J., Kalton, N.J., Kutzarova, Denka, and Temlyakov, V.N. (2003b). The thresholding greedy algorithm, greedy bases, and duality. Constructive Approx., 19 575597.CrossRefGoogle Scholar
Dilworth, S.J., Soto-Bajo, M., and Temlyakov, V.N. (2012). Quasi-greedy bases and Lebesgue-type inequalities. Stud. Math., 211 4169.CrossRefGoogle Scholar
Zung, Dinh [Dung, Dinh] (1984). Approximation of classes of functions on the torus defined by a mixed modulus of continuity. In Constructive Theory of Functions (Proc. Internat. Conf., Varna, 1984). Bulgarian Academy of Science, 4348.Google Scholar
Dung, Dinh (1985). Approximation of multivariate functions by means of harmonic analysis. Dissertation, Moscow, MGU.Google Scholar
Zung, Dinh [Dung, Dinh] (1986). Approximation by trigonometric polynomials of functions of several variables on the torus. Mat. Sb., 131 251271; English translation in Mat. Sb., 59.Google Scholar
Dung, Dinh (1991). On optimal recovery of multivariate periodic functions, In Proc. ICM-90 Satellite Conf. on Harmonic Analysis, Igary, S. (ed). Springer-Verlag, 96105.Google Scholar
Dung, Dinh and Ullrich, T. (2014). Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square. Math. Nachr., 288 743762.CrossRefGoogle Scholar
Dung, Ding, Temlyakov, V.N., and Ullrich, T. (2016). Hyperbolic cross approximation. arXiv:1601.03978v1 [math.NA], accessed 15 Jan 2016.Google Scholar
Donahue, M., Gurvits, L., Darken, C., and Sontag, E. (1997). Rate of convex approximation in non-Hilbert spaces. Constructive Approx., 13 187220.CrossRefGoogle Scholar
Donoho, D., Elad, M., and Temlyakov, V.N. (2007). On the Lebesgue type inequalities for greedy approximation. J. Approximation Theory, 147 185195.CrossRefGoogle Scholar
Dubinin, V.V. (1992). Cubature formulas for classes of functions with bounded mixed difference. Mat. Sb., 183; English translation in Mat. Sb., 76 283292.Google Scholar
Dubinin, V.V. (1997). Greedy algorithms and applications. Ph.D. thesis, University of South Carolina, 1997.Google Scholar
Dzyadyk, V.K. (1977). Introduction to the Theory of Uniform Approximation of Functions by Polynomials. Nauka.Google Scholar
Favard, J. (1937). Sur les meilleurs procédés d’approximation de certaines classes de fonctions par des polynomes trigonometriques. Bull. Sci. Math., 61 209224; 243–256.Google Scholar
Foucart, S. (2012). Sparse recovery algorithms: sufficient conditions in terms of restricted isometry constants. In Proc. Conf. on Approximation Theory XIII: San Antonio, 2010, 65–77.Google Scholar
Franke, J. (1986). On the spaces of Triebel–Lizorkin type: pointwise multipliers and spaces on domains. Math. Nachr. 125 2968.CrossRefGoogle Scholar
Fredholm, I. (1903). Sur une classe d’equations fonctionnelles. Acta Math., 27 365390.CrossRefGoogle Scholar
Frolov, K.K. (1976). Upper bounds on the error of quadrature formulas on classes of functions. Dokl. Akad. Nauk SSSR, 231 818821; English translation in Soviet Math. Dokl., 17.Google Scholar
Frolov, K.K. (1979). Quadrature formulas on classes of functions. PhD dissertation, Vychisl. Tsentr Academy Nauk SSSR.Google Scholar
Frolov, K.K. (1980). An upper estimate of the discrepancy in the Lp-metric, 2 ≤ p < ∞. Dokl. Akad. Nauk SSSR, 252 805807; English translation in Soviet Math. Dokl., 21.Google Scholar
Galeev, E.M. (1978). Approximation of classes of functions with several bounded derivatives by Fourier sums. Matem. Zametki, 23 197212; English translation in Math. Notes, 23.Google Scholar
Galeev, E.M. (1982). Order estimates of derivatives of the multidimensional periodic Dirichlet α-kernel in a mixed norm. Mat. Sb., 117(159) 3243; English translation in Mat. Sb., 45.Google Scholar
Galeev, E.M. (1984). Kolmogorov widths of certain classes of periodic functions of several variables. In Constructive Theory of Functions (Proc. Internat. Conf., Varna, 1984). Publ. House Bulgarian Acad. Sci. 2732.Google Scholar
Galeev, E.M. (1985). Kolmogorov widths of the classes Wp and Hp of periodic functions of several variables in the space Lq. Izv. Akad. Nauk SSSR, 49 916934; English translation in Math. Izv. Acad. Sci. USSR 27 (1986).Google Scholar
Galeev, E.M. (1988). Orders of orthogonal projection widths of classes of periodic functions of one and several variables. Matem. Zametki, 43 197211; English translation in Math. Notes, 43.Google Scholar
Galeev, E.M. (1990). Kolmogorov widths of classes of periodic functions of one and several variables. Izv. Akad. Nauk SSSR, 54 418430; English translation in Math. Izv. Acad. of Sciences USSR, 36 (1991).Google Scholar
Garnaev, A.Yu. and Gluskin, E.D. (1984). On widths of the Euclidean ball. Dokl. Akad. Nauk SSSR, 277 10481052; English translation in Soviet Math. Dokl., 30.Google Scholar
Gao, F., Ing, C-K., and Yang, Y. (2013). Metric entropy and sparse linear approximation of ℓq-hulls for 0 < q ≤ 1. J. Approx. Theory, 166 4255.CrossRefGoogle Scholar
Garrigós, G., Hernández, E., and Oikhberg, T. (2013). Lebesgue type inequalities for quasi-greedy bases. Constr. Approx., 38 447479.CrossRefGoogle Scholar
Gilbert, A.C., Muthukrishnan, S. and Strauss, M.J. (2003). Approximation of functions over redundant dictionaries using coherence. In The 14th Annual ACM–SIAM Symp. on Discrete Algorithms. SIAM, 243252.Google Scholar
Gluskin, E.D. (1974). On a problem concerning widths. Dokl. Akad. Nauk SSSR, 219 (1974), 527530; English translation in Soviet Math. Dokl. 15 (1974).Google Scholar
Gluskin, E.D. (1983). Norms of random matrices and widths of finite-dimensional sets. Mat. Sb., 120 (162) 180189; English translation in Mat. Sb., 48 (1984).Google Scholar
Gluskin, E.D. (1989). Extremal properties of orthogonal parallelpipeds and their application to the geometry of Banach spaces. Mat. Sb., 64 8596.CrossRefGoogle Scholar
Gribonval, R. and Nielsen, M. (2001). Some remarks on non-linear approximation with Schauder bases. East J. Approx., 7 267285.Google Scholar
Griebel, M. (2006). Sparse grids and related approximation schemes for higher dimensional problems. In Proc. Conf. on Foundations of Computational Mathematics, Santander 2005, pp. 106161. London Mathematical Society Lecture Notes Series, vol. 331, Cambridge University Press.CrossRefGoogle Scholar
Györfy, L., Kohler, M., Krzyzak, A., and Walk, H. (2002). A Distribution-Free Theory of Non-Parametric Regression. Springer-Verlag.CrossRefGoogle Scholar
Hadwiger, H. (1957). Vorlesungen über Inhalt, Oberflüsche und Isoperimetrie. Springer-Verlag.CrossRefGoogle Scholar
Halász, G. (1981). On Roth’s method in the theory of irregularities of point distributions. In Proc. Conf. on Recent Progress in Analytic Number Theory Vol. 2 (Durham, 1979), Academic Press, 7994.Google Scholar
Halton, J.H. and Zaremba, S.K. (1969). The extreme and L2 discrepancies of some plane sets. Monats. Math., 73 316328.CrossRefGoogle Scholar
Hardy, G.H. and Littlewood, J.E. (1928). Some properties of fractional integrals. I. Math. Zeit., 27 565– 606.Google Scholar
Hardy, G.H. and Littlewood, J.E. (1966). In Collected Papers of G. Hardy, vol. 1, Clarendon Press, 113114.Google Scholar
Heinrich, S., Novak, E., Wasilkowski, G. and Wozniakowski, H. (2001). The inverse of the star-discrepancy depends linearly on the dimension. Acta Arithmetica, 96 279302.CrossRefGoogle Scholar
Hlawka, E. (1962). Zur angenaherten Berechnung mehrfacher Integrale. Monats. Math., B66 140151.CrossRefGoogle Scholar
Höllig, K. (1980). Diameters of classes of smooth functions. In Quantitative Approximation, Academic Press, 163176.CrossRefGoogle Scholar
Hsiao, C.C., Jawerth, B., Lucier, B.J., and Yu, X. (1994). Near optimal compression of orthogonal wavelet expansions. In Wavelets: Mathematics and Applications, CRC, 425446.Google Scholar
Ismagilov, R.S. (1974). Widths of sets in normed linear spaces and the approximation of functions by trigonometric polynomials. Uspekhi Mat. Nauk, 29 161178; English translation in Russian Math. Surveys, 29 (1974).Google Scholar
Jackson, D. (1911). Über die Genauigkeit der Annaherung stegiger Function durch ganze rationale Functionen gegebenen Grader und trigonometrishe Summen gegebenen Ordmund. Dissertation, Göttingen.Google Scholar
Jackson, D. (1933). Certain problems of closest approximation. Bull. Amer. Math. Soc., 39 889906.CrossRefGoogle Scholar
Jawerth, B. (1977). Some observations on Besov and Lizorkin–Triebel spaces. Math. Scand. 40 94104.CrossRefGoogle Scholar
Kadec, M.I. and Pelczynski, A. (1962). Bases, lacunary sequences, and complemented subspaces in the spaces Lp. Studia Math., 21 161176.CrossRefGoogle Scholar
Kamont, A. and Temlyakov, V.N. (2004). Greedy approximation and the multivariate Haar system. Studia Math., 161(3) 199223.CrossRefGoogle Scholar
Kashin, B.S. (1977). Widths of certain finite-dimensional sets and classes of smooth functions. Izv. AN SSSR, 41 334351; English translation in Math. Izv., 11 (1977).Google Scholar
Kashin, B.S. (1980). On certain properties of the space of trigonometric polynomials with the uniform norm. Trudy Mat. Inst. Steklov, 145 111116; English translation in Proc. Steklov Inst. Math., 145 (1981).Google Scholar
Kashin, B.S. (1981). Widths of Sobolev classes of small-order smoothness. Vestnik Moskov. Univ., Ser. Mat. Mekh., 5 5054; English translation in Moscow Univ. Math. Bull., 5 62–66.Google Scholar
Kashin, B.S. and Temlyakov, V.N. (1994). On best m-terms approximations and the entropy of sets in the space L1. Mat. Zametki, 56 5786; English translation in Math. Notes, 561137–1157.Google Scholar
Kashin, B.S. and Temlyakov, V.N. (1995). Estimate of approximate characteristics for classes of functions with bounded mixed derivative. Math. Notes, 58 13401342.CrossRefGoogle Scholar
Kashin, B.S. and Temlyakov, V.N. (2003). The volume estimates and their applications. East J. Approx., 9 469485.Google Scholar
Kashin, B.S. and Temlyakov, V.N. (2008). On a norm and approximate characteristics of classes of multivariate functions. J. Math. Sci., 155 5780.CrossRefGoogle Scholar
Keng, Hua Loo and Wang, Yuan (1981). Applications of Number Theory to Numerical Analysis. Springer-Verlag.CrossRefGoogle Scholar
Kerkyacharian, G. and Picard, D. (2006). Nonlinear approximation and Muckenhoupt weights. Constructive Approx., 24 123156.CrossRefGoogle Scholar
Kolmogorov, A.N. (1936). Uber die beste Annäherung von Funktionen einer Funktion-klasse. Ann. Math., 37 107111.CrossRefGoogle Scholar
Kolmogorov, A.N. (1985). Selected Papers, Mathematics and Mechanics. Nauka, Moscow.Google Scholar
Konyagin, S.V. and Temlyakov, V.N. (1999). A remark on greedy approximation in Banach spaces. East. J. Approx., 5 365379.Google Scholar
Konyagin, S.V. and Temlyakov, V.N. (2002). Greedy approximation with regard to bases and general minimal systems. Serdica Math. J., 28 305328.Google Scholar
Konyushkov, A.A. (1958). Best approximations by trigonometric polynomials and Fourier coefficients. Mat. Sb., 44 5384.Google Scholar
Korobov, N.M. (1959). On the approximate computation of multiple integrals. Dokl. Akad. Nauk SSSR, 124 12071210.Google Scholar
Korobov, N.M. (1963). Number-Theoretic Methods in Numerical Analysis. Fizmatgis.Google Scholar
Kuelbs, J. and Li, W.V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Functional Analysis, 116 133157.CrossRefGoogle Scholar
Kuipers, L. and Niederreiter, H. (1974). Uniform Distribution of Sequences. Wiley.Google Scholar
Kulanin, E.D. (1985) On widths of functional classes of small smoothness. Dokl. Bulgarian Acad. Sci., 41 16011602.Google Scholar
Kushpel’, A.K. (1989). Estimates of the widths of classes of analytic functions. Ukrainian Math. Jour., 41 567570; English translation in Ukr. Math. Jour., 41 (1989).Google Scholar
Kushpel’, A.K. (1990). Estimation of the widths of classes of smooth functions in the space Lq. Ukrainian Math. Jour., 42 279280; English translation in Ukr. Math. Jour., 42(1990).Google Scholar
Lebesgue, H. (1909). Sur les intégrales singuliéres. Ann. Fac. Sci. Univ. Toulouse (3), 1 25117.CrossRefGoogle Scholar
Lebesgue, H. (1910). Sur la represantation trigonometrique approchée des fonctions satisfaisants une condition de Lipschitz. Bull. Soc. Math. France, 38 184210.CrossRefGoogle Scholar
Lifshits, M.A. and Tsirelson, B.S. (1986). Small deviations of Gaussian fields. Teor. Probab. Appl., 31 557558.Google Scholar
Lindenstrauss, J. and Tzafriri, L. (1979). Classical Banach Spaces I,II. Springer-Verlag.CrossRefGoogle Scholar
Livshitz, E.D. (2012). On the optimality of the orthogonal greedy algorithms for μ-coherent dictionaries. J. Approx. Theory, 164(5) 668681.CrossRefGoogle Scholar
Livshitz, E.D. and Temlyakov, V.N. (2014). Sparse approximation and recovery by greedy algorithms, IEEE Transactions on Information Theory, 60 39894000.CrossRefGoogle Scholar
Maiorov, V.E. (1975). Discretization of the diameter problem, Uspekhi Matem. Nauk, 30 179180.Google Scholar
Maiorov, V.E. (1978). On various widths of the class in the space Lq. Izv. Akad. Nauk SSSR Ser. Mat., 42 773788; English translation in Math. USSR-Izv., 13 (1979).Google Scholar
Maiorov, V.E. (1986). Trigonometric diameters of the Sobolev classes in the space Lq. Math. Notes 40 590597.Google Scholar
Makovoz, Yu.I. (1972). On a method of estimation from below of diameters of sets in Banach spaces. Mat. Sb., 87, 136142; English translation in Mat. Sb., 16 (1972).Google Scholar
Makovoz, Y. (1984). On trigonometric n-widths and their generalizations. J. Approx. Theory, 41 361366.CrossRefGoogle Scholar
Matous̆ek, J. (1999). Geometric Discrepancy. Springer-Verlag.CrossRefGoogle Scholar
Mityagin, B.S. (1962). Approximation of functions in the spaces Lp and C on the torus, Mat. Sb., 58 397414.Google Scholar
Needell, D. and Tropp, J.A. (2009). CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmonic Anal., 26 301321.CrossRefGoogle Scholar
Needell, D. and Vershynin, R. (2009). Uniform uncertainty principle and signal recovery via orthogonal matching pursuit. Found. Comp. Math., 9 317334.CrossRefGoogle Scholar
Niederreiter, H., Tichy, R.F. and Turnwald, G. (1990). An inequality for differences of distribution functions. Arch. Math., 54 166172.CrossRefGoogle Scholar
Nielsen, M. (2007). An example of an almost greedy uniformly bounded orthonormal basis for Lp(0,1). J. Approx. Theory, 149 188192.CrossRefGoogle Scholar
Nikol’skaya, N.S. (1974). Approximation of differentiable functions of several variables by Fourier sums in the Lp-metric., Sibirsk. Mat. Zh., 15 395412; English translation in Siberian Math. J., 15 (1974).Google Scholar
Nikol’skaya, N.S. (1975). Approximation of periodic functions in the class by Fourier sums. Sibirsk. Mat. Zh., 16 761780; English translation in Siberian Math. J., 16 (1975).Google Scholar
Nikol’skii, S.M. (1951). Inequalities for entire functions of exponential type and their use in the theory of differentiable functions of several variables. Trudy MIAN, 38 244278; English translation in Amer. Math. Soc. Transl. (2), 80 (1969).Google Scholar
Nikol’skii, S.M. (1963). Functions with dominating mixed derivative satisfying a multiple Hölder condition. Sibirsk. Mat. Zh., 4 13421364; English translation in Amer. Math. Soc. Transl. (2), 102 (1973).Google Scholar
Nikol’skii, S.M. (1969). Approximation of Functions of Several Variables and Imbedding Theorems. Nauka; English translation published by Springer, 1975.Google Scholar
Nikol’skii, S.M. (1979). Quadrature Formulas. Nauka.Google Scholar
Novak, E. (1988). Deterministic and Stochastic Error Bounds in Numerical Analysis, Lecture Notes in Mathematics, 1349, Springer-Verlag.CrossRefGoogle Scholar
Offin, D. and Oskolkov, K. (1993). A note on orthonormal polynomial bases and wavelets. Constructive Approx., 9 319325.CrossRefGoogle Scholar
Quade, E. (1937). Trigonometric approximation in the mean. Duke Math J., 3 529543.CrossRefGoogle Scholar
Romanyuk, A.S. (2003). Best M-term trigonometric approximations of Besov classes of periodic functions of several variables. Izvestia RAN, Ser. Mat., 67 61100; English translation in Izvestiya Math., 67 265.Google Scholar
Roth, K.F. (1954). On irregularities of distribution. Mathematika, 1 7379.CrossRefGoogle Scholar
Roth, K.F. (1976). On irregularities of distribution. II. Comm. Pure Appl. Math., 29 749754.CrossRefGoogle Scholar
Roth, K.F. (1979). On irregularities of distribution. III. Acta Arith., 35 373384.CrossRefGoogle Scholar
Roth, K.F. (1980). On irregularities of distribution. IV. Acta Arith., 37 6775.CrossRefGoogle Scholar
Rudin, W. (1952). L2-approximation by partial sums of orthogonal developments. Duke Math. J., 19 14.CrossRefGoogle Scholar
Rudin, W. (1959). Some theorems on Fourier coefficients. Proc. Amer. Math. Soc., 10 855859.CrossRefGoogle Scholar
Savu, D. and Temlyakov, V.N. (2013). Lebesgue-type inequalities for greedy approximation in Banach spaces. IEEE Trans. Inform. Theory, 58 10981106.CrossRefGoogle Scholar
Schmidt, W.M. (1972). Irregularities of distribution, VII. Acta Arith., 21 4550.CrossRefGoogle Scholar
Schmidt, W.M. (1977a). Irregularities of distribution, X. In Number Theory and Algebra. Academic Press, 311329.Google Scholar
Schmidt, W.M. (1977b). Lectures on Irregularities of Distribution. Tata Institute of Fundamental Research.Google Scholar
Schütt, C. (1984). Entropy numbers of diagonal operators between symmetric Banach spaces. J. Approx. Theory, 40 121128.CrossRefGoogle Scholar
Shapiro, H.S. (1951). Extremal problems for polynomials and power series. M.S. thesis, MIT. Massachusetts Institute of Technology.Google Scholar
Skriganov, M.M. (1994). Constructions of uniform distributions in terms of geometry of numbers. Algebra Anal., 6 200230.Google Scholar
Smolyak, S.A. (1960). The ε-entropy of the classes and in the metric L2. Dokl. Akad. Nauk SSSR, 131 3033.Google Scholar
Smolyak, S.A. (1963). Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR, 148 10421045; English translation in Soviet Math. Dokl., 4 (1963).Google Scholar
Sobolev, S.L. (1994). Introduction to the Theory of Cubature Formulas. Nauka.Google Scholar
Stechkin, S.B. (1951). On the degree of best approximation of continuous functions. Izvestia AN SSSR, Ser. Mat., 15 219242.Google Scholar
Stechkin, S.B. (1954). On the best approximation of given classes of functions by arbitrary polynomials. Uspekhi Mat. Nauk, 9 133134 (in Russian).Google Scholar
Stechkin, S.B. (1955). On absolute convergence of orthogonal series. Dokl. AN SSSR, 102 3740 (in Russian).Google Scholar
Stepanets, A.I. (1987). Classification and Approximation of Periodic Functions. Naukova Dumka.Google Scholar
Talagrand, M. (1994). The small ball problem for the Brownian sheet. Ann. Probab., 22 13311354.CrossRefGoogle Scholar
Telyakovskii, S.A. (1963). On estimates of the derivatives of trigonometric polynomials in several variables. Siberian Math. Zh., 4 14041411.Google Scholar
Telyakovskii, S.A. (1964). Some estimates for trigonometric series with quasi-convex coefficients, Mat. Sb., 63 426444; English translation in Amer. Math. Soc. Transl. 86.Google Scholar
Telyakovskii, S.A. (1988). Research in the theory of approximation of functions at the Mathematical Institute of the Academy of Sciences, Trudy MIAN, 182 128179; English translation in Proc. Steklov Inst. Math. 1 (1990).Google Scholar
Temlyakov, V.N. (1979). Approximation of periodic functions of several variables with bounded mixed derivative. Dokl. Akad. Nauk SSSR, 248 527531; English translation in Soviet Math. Dokl., 20 (1979).Google Scholar
Temlyakov, V.N. (1980a). On the approximation of periodic functions of several variables with bounded mixed difference. Dokl. Akad. Nauk SSSR, 253 544548; English translation in Soviet Math. Dokl., 22 (1980).Google Scholar
Temlyakov, V.N. (1980b). Approximation of periodic functions of several variables with bounded mixed difference. Mat. Sb., 133 6585; English translation in Math. USSR Sbornik 41 (1982).Google Scholar
Temlyakov, V.N. (1980c). Approximation of periodic functions of several variables with bounded mixed derivative. Trudy MIAN, 156 233260; English translation in Proc. Steklov Inst. Math., 2 (1983).Google Scholar
Temlyakov, V.N. (1982a). Widths of some classes of functions of several variables. Dokl. Akad. Nauk SSSR, 267 314317; English translation in Soviet Math. Dokl., 26.Google Scholar
Temlyakov, V.N. (1982b). Approximation of functions with a bounded mixed difference by trigonometric polynomials and the widths of some classes of functions. Izv. Akad. Nauk SSSR, 46 171186; English translation in Math. Izv. Acad. Sci. USSR, 20 (1983).Google Scholar
Temlyakov, V.N. (1985a). On the approximate reconstruction of periodic functions of several variables. Dokl. Akad. Nauk SSSR, 280 13101313; English translation in Soviet Math. Dokl., 31.Google Scholar
Temlyakov, V.N. (1985b). Approximate recovery of periodic functions of several variables. Mat. Sb., 128 256268; English translation in Mat. Sb., 56 (1987).Google Scholar
Temlyakov, V.N. (1985c). Quadrature formulas and recovery on the values at the knots of number-theoretical nets for classes of functions of small smoothness. Uspekhi Matem. Nauk, 40 203204.Google Scholar
Temlyakov, V.N. (1985d). Approximation of periodic functions of several variables by trigonometric polynomials, and widths of some classes of functions. Izv. Akad. Nauk SSSR, 49 9861030; English translation in Math. USSR Izv., 27 (1986).Google Scholar
Temlyakov, V.N. (1985e). On linear bounded methods of approximation of functions. Dokl. Sem. Inst. Prikl. Mat. Vekua, 1 144147.Google Scholar
Temlyakov, V.N. (1986a). On reconstruction of multivariate periodic functions based on their values at the knots of number-theoretical nets. Anal. Math., 12 287305.Google Scholar
Temlyakov, V.N. (1986b). Approximation of periodic functions of several variables by bilinear forms. Izvestiya AN SSSR, 50 137155; English translation in Math. USSR Izvestiya, 28 133–150.Google Scholar
Temlyakov, V.N. (1986c). Approximation of functions with bounded mixed derivative. Trudy MIAN, 178 1112; English translation in Proc. Steklov Inst. Math., 1 (1989).Google Scholar
Temlyakov, V.N. (1987). Estimates of the best bilinear approximations of functions of two variables and some of their applications, Mat. Sb., 134 93107; English translation in Math. USSR – Sb 62 (1989), 95–109.Google Scholar
Temlyakov, V.N. (1988a). On estimates of ε-entropy and widths of classes of functions with bounded mixed derivative or difference. Dokl. Akad. Nauk SSSR, 301 288291; English translation in Soviet Math. Dokl., 38, 84–87.Google Scholar
Temlyakov, V.N. (1988b). Estimates of best bilinear approximations of periodic functions. Trudy Mat. Inst. Steklov, 181 250267; English translation in Proc. Steklov Inst. Math., 4(1989), 275–293.Google Scholar
Temlyakov, V.N. (1988c). Approximation by elements of a finite-dimensional subspace of functions from various Sobolev or Nikol’skii spaces. Matem. Zametki, 43 770786; English translation in Math. Notes, 43.Google Scholar
Temlyakov, V.N. (1989a). Error estimates of quadrature formulas for classes of functions with bounded mixed derivative. Matem. Zametki, 46 128134; English translation in Math. Notes, 46.Google Scholar
Temlyakov, V.N. (1989b). Approximation of functions of several variables by trigonometric polynomials with harmonics from hyperbolic crosses. Ukrainian Math. J., 41 518524; English translation in Ukr. Math. J., 41.CrossRefGoogle Scholar
Temlyakov, V.N. (1989c). Bilinear approximation and applications. Trudy Mat. Inst. Steklov, 187 191215; English translation in Proc. Steklov Inst. Math., 3 (1990), 221–248.Google Scholar
Temlyakov, V.N. (1989d). Estimates of the asymptotic characteristics of classes of functions with bounded mixed derivative or difference. Trudy Matem. Inst. Steklov, 189 138168; English translation in Proc. Steklov Inst. Math., 4 161–197.Google Scholar
Temlyakov, V.N. (1990a). On a problem of estimating widths of classes of infinity differentiable functions. Matem. Zametki 47 155157.Google Scholar
Temlyakov, V.N. (1990b) On a way of obtaining lower estimates for the errors of quadrature formulas. Matem. Sbornik, 181 (1990), 14031413; English translation in Math. USSR Sbornik, 71 (1992).Google Scholar
Temlyakov, V.N. (1991a). On universal cubature formulas. Dokl. Akad. Nauk SSSR, 316 3447; English translation in Soviet Math. Dokl., 43 (1991), 39–42.Google Scholar
Temlyakov, V.N. (1991b). Error estimates for Fibonacci quadrature formulas for classes of functions with bounded mixed derivative. Trudy MIAN, 200 327335; English translation in Proc. Steklov Inst. Math., 2 (1993).Google Scholar
Temlyakov, V.N. (1992a). Bilinear approximation and related questions. Trudy Mat. Inst. Steklov, 194 229248; English translation in Proc. Steklov Inst. Math., 4 (1993), 245–265.Google Scholar
Temlyakov, V.N. (1992b). Estimates of best bilinear approximations of functions and approximation numbers of integral operators. Mat. Zametki, 51 125134; English translation in Math. Notes, 51, 510–517.Google Scholar
Temlyakov, V.N. (1993a). On approximate recovery of functions with bounded mixed derivative. J. Complexity, 9 4159.CrossRefGoogle Scholar
Temlyakov, V.N. (1993b) Approximation of Periodic Functions. Nova Science Publishers.Google Scholar
Temlyakov, V.N. (1994). On error estimates for cubature formulas. Trudy Matem. Inst. Steklova, 207 326338; English translation in Proc. Steklov Inst. Math., 6, (1995).Google Scholar
Temlyakov, V.N. (1995a). An inequality for trigonometric polynomials and its application for estimating the entropy numbers. J. Complexity, 11 293307.CrossRefGoogle Scholar
Temlyakov, V.N. (1995b). Some inequalities for multivariate Haar polynomials. East J. Approx., 1 6172.Google Scholar
Temlyakov, V.N. (1996). An inequality for trigonometric polynomials and its application for estimating the Kolmogorov widths. East J. Approx., 2 253262.Google Scholar
Temlyakov, V.N. (1998a). On two problems in the multivariate approximation. East J. Approx., 4 505514.Google Scholar
Temlyakov, V.N. (1998b). Nonlinear Kolmogorov’s widths. Matem. Zametki, 63, 891902.Google Scholar
Temlyakov, V.N. (1998c). Greedy algorithm and m-term trigonometric approximation. Constr. Approx., 14, 569587.CrossRefGoogle Scholar
Temlyakov, V.N. (1998d). Nonlinear m-term approximation with regard to the multivariate Haar system. East J. Approx., 4 87106.Google Scholar
Temlyakov, V.N. (1998e). The best m-term approximation and greedy algorithms, Advances in Comp. Math., 8 249265.CrossRefGoogle Scholar
Temlyakov, V.N. (2000a). Weak greedy algorithms, Advances in Comput. Math., 12 213227.CrossRefGoogle Scholar
Temlyakov, V.N. (2000b). Greedy algorithms with regards to multivariate systems with special structure. Constr. Approx., 16 399425.CrossRefGoogle Scholar
Temlyakov, V.N. (2001). Greedy algorithms in Banach spaces. Adv. Comput. Math., 14 277292.CrossRefGoogle Scholar
Temlyakov, V.N. (2002a). Universal bases and greedy algorithms for anisotropic function classes. Constr. Approx., 18 529550.CrossRefGoogle Scholar
Temlyakov, V.N. (2002b). Nonlinear approximation with regard to bases. In Approximation Theory X. Vanderbilt University Press 373402.Google Scholar
Temlyakov, V.N. (2003a). Nonlinear method of approximation. Found. Compt. Math., 3 33107.CrossRefGoogle Scholar
Temlyakov, V.N. (2003b). Cubature formulas and related questions, J. Complexity, 19 352391.CrossRefGoogle Scholar
Temlyakov, V.N. (2005). Greedy-type approximation in Banach spaces and applications. Constr. Approx., 21 257292.CrossRefGoogle Scholar
Temlyakov, V.N. (2006). On universal estimators in learning theory. Trudy MIAN im. VA Steklova, 255 256272; English translation in Proc. Steklov Inst. Math., 255 (2006), 244–259.Google Scholar
Temlyakov, V.N. (2007). Greedy approximation in Banach spaces. In Banach Spaces and their Applications in Analysis. de Gruyter, 193208.Google Scholar
Temlyakov, V.N. (2008). Greedy approximation. Acta Numerica, 17 235409.CrossRefGoogle Scholar
Temlyakov, V.N. (2011). Greedy Approximation. Cambridge University Press.CrossRefGoogle Scholar
Temlyakov, V.N. (2013). An inequality for the entropy numbers and its application. J. Approx. Theory, 173 110121.CrossRefGoogle Scholar
Temlyakov, V.N. (2014). Sparse approximation and recovery by greedy algorithms in Banach spaces. Forum of Mathematics, Sigma, 2 e12, 26 pp.CrossRefGoogle Scholar
Temlyakov, V.N. (2015a). Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness. Matem. Sb., 206, 131160. ArXiv: 1412.8647v1 [math.NA] 24 December 2014, 1–37.Google Scholar
Temlyakov, V.N. (2015b). Constructive sparse trigonometric approximation for functions with small mixed smoothness. ArXiv: 1503.00282v1 [math.NA] 1 March 2015, 1–30.Google Scholar
Temlyakov, V.N. (2015c). Sparse approximation with bases. In Proc. Conf. on Advanced Courses in Mathematics CRM Barcelona, Birkhäuser–Springer.Google Scholar
Temlyakov, V.N. (2016a). Incremental greedy algorithm and its applications in numerical integration. In Proc. Conf. on Monte Carlo and Quasi-Monte Carlo Methods, Leuven, April 2014. Springer Proceedings in Mathematics and Statistics, 163 557570.CrossRefGoogle Scholar
Temlyakov, V.N. (2016b) On the entropy numbers of the mixed smoothness function classes. ArXiv:1602.08712v1 [math.NA] 28 February 2016.Google Scholar
Temlyakov, V.N and Zheltov, P. (2011). On performance of greedy algorithms. J. Approx. Theory, 163 11341145.CrossRefGoogle Scholar
Temlyakov, V.N., Yang, Mingrui and Ye, Peixin (2011). Greedy approximation with regard to non-greedy bases. Adv. Comput. Math., 34 319337.CrossRefGoogle Scholar
Tikhomirov, V.M. (1960a). On n-dimensional diameters of certain functional classes. Dokl. Akad. Nauk SSSR, 130 734737; English translation in Soviet Math. Dokl., 1.Google Scholar
Tikhomirov, V.M. (1960b). Widths of sets in function spaces and the theory of best approximation. Uspekhi Matem. Nauk, 15 81120; English translation in Russian Math. Surveys, 15.Google Scholar
Tikhomirov, V.M. (1976). Some Topics in Approximation Theory. Moscow State University.Google Scholar
Timan, A.F. (1960). Theory of Approximation of Functions of a Real Variable. Phys.–Math. Lit., Moscow, 1960; English translation published by MacMillan, 1963.Google Scholar
Timan, M.F. (1974). On embeddings of the function classes . Izv. Vyssh. Uchebn. Zaved. Mat., 10 6174; English translation in Soviet Math. Iz. VUZ., 18.Google Scholar
Triebel, H. (2010). Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration. European Mathematical Society.CrossRefGoogle Scholar
Triebel, H. (2015). Global solutions of Navier–Stokes equations for large initial data belonging to spaces with dominating mixed smoothness. J. Complexity, 31 147161.CrossRefGoogle Scholar
Trigub, R.M. (1971). Summability and absolute convergence of the Fourier series in total. In Metric Questions of Theory of Approximation and Mapping. Naukova Dumka 173266.Google Scholar
Trigub, R.M. and Belinsky, E.S. (2004). Fourier Analysis and Approximation of Functions. Kluwer Academic Publishers.CrossRefGoogle Scholar
Tropp, J.A. (2004). Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inform. Theory, 50 22312242.CrossRefGoogle Scholar
Ul’yanov, P.L. (1970). Embedding theorems and relations between best approximations (moduli of continuity) in different metrics, Mat. Sb., 81 (123) (1970), 104131; English translation in Mat. Sb., 10 (1970).Google Scholar
Uninskii, A.P. (1966). Inequalities in the mixed norm for the trigonometric polynomials and entire functions of finite degree. In Mater. Vsesoyuzn. Simp. Teor. Vlozhen., Baku.Google Scholar
van Aardenne-Ehrenfest, T. (1945). Proof of the impossibility of a just distribution of an infinite sequence of points over an interval. Proc. Kon. Ned. Akad. v. Wetensch, 48 266271.Google Scholar
van der Corput, J.G. (1935a). Verteilungsfunktionen. I. Proc. Kon. Ned. Akad. v. Wetensch., 38 813821.Google Scholar
van der Corput, J.G. (1935b). Verteilungsfunktionen. II, Proc. Kon. Ned. Akad. v. Wetensch., 38 10581066.Google Scholar
Vilenkin, I.V. (1967). Plane nets of integration. Zhur. Vychisl. Mat. i Mat. Fis., 7, 189196; English translation in USSR Comp. Math. and Math. Phys., 7, 258–267.Google Scholar
Wang, J. and Shim, B. (2012). Improved recovery bounds of orthogonal matching pursuit using restricted isometry property. ArXiv:1211.4293v1 [cs.IT] 19 Nov 2012.Google Scholar
Wojtaszczyk, P. (2000). Greedy algorithm for general biorthogonal systems. J. Approx. Theory 107 293314.CrossRefGoogle Scholar
Yserentant, H. (2010). Regularity and Approximability of Electronic Wave Functions. Lecture Notes in Mathematics, Springer.CrossRefGoogle Scholar
Zhang, T. (2011). Sparse recovery with orthogonal matching pursuit under RIP, IEEE Transactions on Information Theory, 57 62156221.CrossRefGoogle Scholar
Zygmund, A. (1959). Trigonometric Series. Cambridge University Press.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • V. Temlyakov, University of South Carolina
  • Book: Multivariate Approximation
  • Online publication: 06 July 2018
  • Chapter DOI: https://doi.org/10.1017/9781108689687.012
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • V. Temlyakov, University of South Carolina
  • Book: Multivariate Approximation
  • Online publication: 06 July 2018
  • Chapter DOI: https://doi.org/10.1017/9781108689687.012
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • V. Temlyakov, University of South Carolina
  • Book: Multivariate Approximation
  • Online publication: 06 July 2018
  • Chapter DOI: https://doi.org/10.1017/9781108689687.012
Available formats
×