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  • Cited by 147
Publisher:
Cambridge University Press
Online publication date:
August 2012
Print publication year:
2011
Online ISBN:
9780511762291

Book description

This first book on greedy approximation gives a systematic presentation of the fundamental results. It also contains an introduction to two hot topics in numerical mathematics: learning theory and compressed sensing. Nonlinear approximation is becoming increasingly important, especially since two types are frequently employed in applications: adaptive methods are used in PDE solvers, while m-term approximation is used in image/signal/data processing, as well as in the design of neural networks. The fundamental question of nonlinear approximation is how to devise good constructive methods (algorithms) and recent results have established that greedy type algorithms may be the solution. The author has drawn on his own teaching experience to write a book ideally suited to graduate courses. The reader does not require a broad background to understand the material. Important open problems are included to give students and professionals alike ideas for further research.

Reviews

'The author is the leading expert on greedy approximation and this book offers a guided tour through the state of the art of the subject. Temlyakov's book is an excellent mathematical monograph and a valuable reference for researchers not only in approximation theory, but also in numerical mathematics, analysis, functional analysis, and statistics. The book is addressed mainly to researchers interested in greedy approximation and related areas. However, it is written at a level that is approachable for graduate students interested in the aforementioned areas, and it could be used for designing graduate courses in greedy approximation, learning theory and compressed sensing. As an added bonus, the author has included an extensive list of open problems in the area that can serve as inspiration for future research papers and dissertations.'

Morten Nielsen Source: SIAM News

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Contents

References
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, 266–271.
Alon, N. (2003), Problems and results in extremal combinatorics, Discrete Math., 273, 31–53.
Baishanski, B. M. (1983), Approximation by polynomials of given length, Illinois J. Math., 27, 449–458.
Bakhvalov, N. S. (1959), On the approximate computation of multiple integrals, Vestnik Moskov. Univ. Ser. Mat. Mekh. Astr. Fiz. Khim., 4, 3–18.
Bakhvalov, N. S. (1963), 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, 5–63.
Baraniuk, R., Davenport, M., Devore, R. and Wakin, M. (2008), A simple proof of the Restricted Isometry Property for random matrices, Construct. Approx., 28, 253–263.
Barron, A. R. (1991), Complexity regularization with applications to artificial neural networks. In Nonparametric Functional Estimation, G., Roussas, ed. (Dordrecht: Kluwer), pp. 561–576.
Barron, A. R. (1993), Universal approximation bounds for superposition of n sigmoidal functions, IEEE Trans. Inf. Theory, 39, 930–945.
Barron, A., Birgé, L. and Massart, P. (1999), Risk bounds for model selection via penalization, Prob. Theory Related Fields, 113, 301–413.
Barron, A., Cohen, A., Dahmen, W. and Devore, R. (2008), Approximation and learning by greedy algorithms, Ann. Stat., 36, 64–94.
Bass, R. F. (1988), Probability estimates for multiparameter Brownian processes, Ann. Prob., 16, 251–264.
Bary, N. K. (1961), Trigonometric Series, (Moscow: Nauka) (in Russian); English translation (Oxford: Pergamon Press, 1964).
Beck, J. and Chen, W. (1987), Irregularities of Distribution, (Cambridge: Cambridge University Press).
Bednorz, W. (2008), Greedy bases are best for m-term approximation, Construct. Approx., 28, 265–275.
Belinsky, E. S. (1998), Estimates of entropy numbers and Gaussian measures for classes of functions with bounded mixed derivative, J. Approx. Theory, 93, 114–127.
Bilyk, D. and Lacey, M. (2008), On the small ball inequality in three dimensions, Duke Math J., 143, 81–115.
Bilyk, D., Lacey, M. and Vagharshakyan, A. (2008), On the small ball inequality in all dimensions, J. Func. Anal., 254, 2470–2502.
Binev, P., Cohen, A., Dahmen, W., Devore, R. and Temlyakov, V. (2005), Universal algorithms for learning theory. Part I: Piecewise constant functions, J. Machine Learning Theory (JMLT), 6, 1297–1321.
Bourgain, J. (1992), A remark on the behaviour of Lp-multipliers and the range of operators acting on Lp-spaces, Israel J. Math., 79, 193–206.
Bykovskii, V. A. (1985), On the correct order of the error of optimal cubature formulas in spaces with dominant derivative, and on quadratic deviations of grids, Preprint, Computing Center Far-Eastern Scientific Center, Akad. Sci. USSR, Vladivostok.
Candes, E. (2006), Compressive sampling, ICM Proc., Madrid, 3, 1433–1452.
Candes, E. (2008), The restricted isometry property and its implications for compressed sensing, C. R. Acad. Sci. Paris, Ser. I 346, 589–592.
Candes, E., Romberg, J. and Tao, T. (2006), Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 59, 1207–1223.
Candes, E. and Tao, T. (2005), Decoding by linear programming, IEEE Trans. Inform. Theory, 51, 4203–4215.
Carl, B. (1981), Entropy numbers, s-numbers, and eigenvalue problems, J. Func. Anal., 41, 290–306.
Carlitz, L. and Uchiyama, S. (1957), Bounds for exponential sums, Duke Math. J., 24, 37–41.
Chazelle, B. (2000), The Discrepancy Method, (Cambridge: Cambridge University Press).
Chen, W. W. L. (1980), On irregularities of distribution, Mathematika, 27, 153–170.
Chen, S. S., Donoho, D. L. and Saunders, M. A. (2001), Atomic decomposition by basis pursuit, SIAM Rev., 43, 129–159.
Cohen, A., Dahmen, W. and Devore, R. (2007), A taste of compressed sensing, Proc. SPIE, Orlando, March 2007.
Cohen, A., Dahmen, W. and Devore, R. (2009), Compressed sensing and k-term approximation, J. Amer. Math. Soc., 22, 211–231.
Cohen, A., Devore, R. A. and Hochmuth, R. (2000), Restricted nonlinear approximation, Construct. Approx., 16, 85–113.
Coifman, R. R. and Wickerhauser, M. V. (1992), Entropy-based algorithms for best-basis selection, IEEE Trans. Inform. Theory, 38, 713–718.
Conway, J. H., Hardin, R. H. and Sloane, N. J. A. (1996), Packing lines, planes, etc.: packing in Grassmannian spaces, Experiment. Math. 5, 139–159.
Conway, J. H. and Sloane, N. J. A. (1998), Sphere Packing, Lattices and Groups (New York: Springer-Verlag).
Cordoba, A. and Fernandez, P. (1998), Convergence and divergence of decreasing rearranged Fourier series, SIAM J. Math. Anal., 29, 1129–1139.
Van Der Corput, J. G. (1935a), Verteilungsfunktionen. I, Proc. Kon. Ned. Akad.v. Wetensch., 38, 813–821.
Van Der Corput, J. G. (1935b), Verteilungsfunktionen. II, Proc. Kon. Ned. Akad.v. Wetensch., 38, 1058–1066.
Cucker, F. and Smale, S. (2001), On the mathematical foundations of learning, Bull. AMS, 39, 1–49.
Dai, W. and Milenkovich, O. (2009), Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inform. Theory, 55, 2230–2249.
Davenport, H. (1956), Note on irregularities of distribution, Mathematika, 3, 131–135.
Davis, G., Mallat, S. and Avellaneda, M. (1997), Adaptive greedy approximations, Construct. Approx., 13, 57–98.
Devore, R. A. (1998), Nonlinear approximation, Acta Numerica, 7, 51–150.
Devore, R. A. (2006), Optimal computation, ICM Proc., Madrid, 1, 187–215.
Devore, R. A. (2007), Deterministic constructions of compressed sensing matricesJ. Complex., 23, 918–925.
Devore, R. A., Jawerth, B. and Popov, V. (1992), Compression of wavelet decompositions, Amer. J. Math., 114, 737–785.
Devore, R. A., Konyagin, S. V. and Temlyakov, V. N. (1998), Hyperbolic wavelet approximation, Construct. Approx., 14, 1–26.
Devore, R. A. and Lorenz, G. G. (1993), Constructive Approximation (Berlin: Springer-Verlag).
Devore, R. A., Petrova, G. and Temlyakov, V. N. (2003), Best basis selection for approximation in Lp, Found. Comput. Math., 3, 161–185.
Devore, R. A. and Popov, V. A. (1988), Interpolation Spaces and Non-linear Approximation, Lecture Notes in Mathematics 1302 (Berlin: Springer), pp. 191–205.
Devore, R. A. and Temlyakov, V. N. (1995), Nonlinear approximation by trigonometric sums, J. Fourier Anal. Appl., 2, 29–48.
Devore, R. A. and Temlyakov, V. N. (1996), Some remarks on Greedy Algorithms, Adv. Comp. Math., 5, 173–187.
Devore, R. A. and Temlyakov, V. N. (1997), Nonlinear approximation in finite-dimensional spaces, J. Complexity, 13, 489–508.
Devore, R. A., Kerkyacharian, G., Picard, D. and Temlyakov, V. (2004), On mathematical methods of learning, IMI Preprints, 10, 1–24.
Devore, R. A., Kerkyacharian, G., Picard, D. and Temlyakov, V. (2006), Mathematical methods for supervised learning, Found. Comput. Math., 6, 3–58.
Dilworth, S. J., Kalton, N. J. and Kutzarova, D. (2003), On the existence of almost greedy bases in Banach spaces, Studia Math., 158, 67–101.
Dilworth, S. J., Kutzarova, D. and Temlyakov, V. (2002), Convergence of some Greedy Algorithms in Banach spaces, J. Fourier Anal. Applic., 8, 489–505.
Dilworth, S. J., Kutzarova, D. and Wojtaszczyk, P. (2002), On approximate ℓ1 systems in Banach spaces, J. Approx. Theory, 114, 214–241.
Dilworth, S. J., Kalton, N. J., Kutzarova, D. and Temlyakov, V. N. (2003), The Thresholding Greedy Algorithm, greedy bases, and duality, Construct. Approx., 19, 575–597.
Ding, Dung (1985), Approximation of multivariate functions by means of harmonic analysis, Hab. Dissertation, Moscow, MGU.
Donahue, M., Gurvits, L., Darken, C. and Sontag, E. (1997), Rate of convex approximation in non-Hilbert spaces, Construct. Approx., 13, 187–220.
Donoho, D. L. (1993), Unconditional bases are optimal bases for data compression and for statistical estimation, Appl. Comput. Harmon. Anal., 1, 100–115.
Donoho, D. L. (1997), CART and Best-Ortho-Basis: a connection, Ann. Stat., 25, 1870–1911.
Donoho, D. L. (2001), Sparse components of images and optimal atomic decompositions, Construct. Approx., 17, 353–382.
Donoho, D. L. (2006), Compressed sensing, IEEE Trans. Inform. Theory, 52, 1289–1306.
Donoho, D. L., Elad, M. and Temlyakov, V. N. (2006), Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Trans. Inf. Theory, 52, 6–18.
Donoho, D. L., Elad, M. and Temlyakov, V. N. (2007), On the Lebesgue type inequalities for greedy approximation, J. Approx. Theory, 147, 185–195.
Donoho, D. L. and Johnstone, I. (1994), Ideal spatial adaptation via wavelet shrinkage, Biometrica, 81, 425–455.
Dubinin, V. V. (1997), Greedy algorithms and applications, Ph.D. Thesis, University of South Carolina.
Dudley, R. M. (1967), The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Func. Anal., 1, 290–330.
Fefferman, C. and Stein, E. (1972), Hp spaces of several variables, Acta Math., 129, 137–193.
Figiel, T., Johnson, W. B. and Schechtman, G. (1988), Factorization of natural embeddings of into Lr, I, Studia Mathematica, 89, 79–103.
Frazier, M. and Jawerth, B. (1990), A discrete transform and decomposition of distribution spaces, J. Funct. Anal., 93, 34–170.
Friedman, J. H. and Stuetzle, W. (1981), Projection pursuit regression, J. Amer. Stat. Assoc., 76, 817–823.
Frolov, K. K. (1976), Upper bounds on the error of quadrature formulas on classes of functions, Dokl. Akad. Nauk SSSR, 231, 818–821; English translation in Sov. Math. Dokl., 17.
Frolov, K. K. (1979), Quadrature formulas on classes of functions, Candidate dissertation, Vychisl. Tsentr Acad. Nauk SSSR, Moscow.
Frolov, K. K. (1980), An upper estimate of the discrepancy in the Lp-metric, 2 ≤ p < ∞, Dokl. Akad. Nauk SSSR, 252, 805–807; English translation in Sov. Math. Dokl., 21.
Galatenko, V. V. and Livshitz, E. D. (2003), On convergence of approximate weak greedy algorithms, East J. Approx., 9, 43–49.
Galatenko, V. V. and Livshitz, E. D. (2005), Generalized approximate weak greedy algorithms, Math. Notes, 78, 170–184.
Ganichev, M. and Kalton, N. J. (2003), Convergence of the Weak Dual Greedy Algorithm in Lp-spaces, J. Approx. Theory, 124, 89–95.
Garnaev, A. and Gluskin, E. (1984), The widths of a Euclidean ball, Dokl. Akad. Nauk USSR, 277, 1048–1052; English translation in Sov. Math. Dokl., 30, 200–204.
Gilbert, A. C., Muthukrishnan, S. and Strauss, M. J. (2003), Approximation of functions over redundant dictionaries using coherence, in M., Farach-Cotton, ed., Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (Philadelphia, PA: SIAM).
Gine, E. and Zinn, J. (1984), Some limit theorems for empirical processes, Ann. Prob., 12, 929–989.
Gluskin, E. D. (1986), An octahedron is poorly approximated by random subspaces, Functsional. Anal. i Prilozhen., 20, 14–20.
Gogyan, S. (2005), Greedy algorithm with regard to Haar subsystems, East J. Approx., 11, 221–236.
Gogyan, S. (2009), On convergence of Weak Thresholding Greedy Algorithm in L1(0, 1), J. Approx. Theory, 161, 49–64.
Gribonval, R. and Nielsen, M. (2001a), Approximate Weak Greedy Algorithms, Adv. Comput. Math., 14, 361–368.
Gribonval, R. and Nielsen, M. (2001b), Some remarks on non-linear approximation with Schauder bases, East J. Approx. 7, 267–285.
Györfy, L., Kohler, M., Krzyzak, A. and Walk, H. (2002), A Distribution-Free Theory of Nonparametric Regression (Berlin: Springer).
Habala, P., Hájek, P. and Zizler, V. (1996), Introduction to Banach spaces [I] (Karlovy: Matfyzpress).
Halász, G. (1981), On Roth's method in the theory of irregularities of points distributions, Recent Prog. Analytic Number Theory, 2, 79–94.
Halton, J. H. and Zaremba, S. K. (1969), The extreme and L2 discrepancies of some plane sets, Monats. für Math., 73, 316–328.
Heinrich, S., Novak, E., Wasilkowski, G. and Wozniakowski, H. (2001), The inverse of the star-discrepancy depends linearly on the dimension, Acta Arithmetica, 96, 279–302.
Hitczenko, P. and Kwapien, S. (1994), On Rademacher series, Prog. Prob., 35, 31–36.
Höllig, K. (1980), Diameters of classes of smooth functions, in R., Devore and K., Scherer, eds., Quantitative Approximation (New York: Academic Press), pp. 163–176.
Huber, P. J. (1985), Projection pursuit, Ann. Stat., 13, 435–475.
Jones, L. (1987), On a conjecture of Huber concerning the convergence of projection pursuit regression, Ann. Stat., 15, 880–882.
Jones, L. (1992), A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training, Ann. Stat., 20, 608–613.
Kalton, N. J., Beck, N. T. and Roberts, J. W. (1984), An F-space Sampler, London Math. Soc. Lecture Notes 5 (Cambridge: Cambridge University Press).
Kamont, A. and Temlyakov, V. N. (2004), Greedy approximation and the multivariate Haar system, Studia Mathematica, 161 (3), 199–223.
Kashin, B. S. (1975), On widths of octahedrons, Uspekhi Matem. Nauk, 30, 251–252.
Kashin, B. S. (1977a), Widths of certain finite-dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR, Ser. Mat., 41, 334–351; English translation in Math. USSR IZV., 11.
Kashin, B. S. (1977b), On the coefficients of expansion of functions from a certain class with respect to complete systems, Siberian J. Math., 18, 122–131.
Kashin, B. S. (1980), On certain properties of the space of trigonometric polynomials with the uniform norm, Trudy Mat. Inst. Steklov, 145, 111–116; English translation in Proc. Steklov Inst. Math. (1981), Issue 1.
Kashin, B. S. (1985), On approximation properties of complete orthonormal systems, Trudy Mat. Inst. Steklov, 172, 187–191; English translation in Proc. Steklov Inst. Math., 3, 207–211.
Kashin, B. S. (2002), On lower estimates for n-term approximation in Hilbert spaces, in B., Bojanov, ed., Approximation Theory: A Volume Dedicated to Blagovest Sendov, (Sofia: DARBA), pp. 241–257.
Kashin, B. S. and Saakyan, A. A. (1989), Orthogonal Series (Providence, RI: American Mathematical Society).
Kashin, B. S. and Temlyakov, V. N. (1994), On best m-term approximations and the entropy of sets in the space L 1, Math. Notes, 56, 1137–1157.
Kashin, B. S. and Temlyakov, V. N. (1995), Estimate of approximate characteristics for classes of functions with bounded mixed derivative, Math. Notes, 58, 1340–1342.
Kashin, B. S. and Temlyakov, V. N. (2003), The volume estimates and their applications, East J. Approx., 9, 469–485.
Kashin, B. S. and Temlyakov, V. N. (2007), A remark on compressed sensing, Math. Notes, 82, 748–755.
Kashin, B. S. and Temlyakov, V. N. (2008), On a norm and approximate characteristics of classes of multivariate functions, J. Math. Sci., 155, 57–80.
Kerkyacharian, G. and Picard, D. (2004), Entropy, universal coding, approximation, and bases properties, Construct. Approx., 20, 1–37.
Kerkyacharian, G., Picard, D. and Temlyakov, V. N. (2006), Some inequalities for the tensor product of greedy bases and weight-greedy bases, East J. Approx., 12, 103–118.
Konyagin, S. V. and Skopina, M. A. (2001), Comparison of the L1-norms of total and truncated exponential sums, Mat. Zametki, 69, 699–707.
Konyagin, S. V. and Temlyakov, V. N. (1999a), A remark on greedy approximation in Banach spaces, East J. Approx., 5, 365–379.
Konyagin, S. V. and Temlyakov, V. N. (1999b), Rate of convergence of Pure Greedy Algorithm, East J. Approx., 5, 493–499.
Konyagin, S. V. and Temlyakov, V. N. (2002), Greedy approximation with regard to bases and general minimal systems, Serdica Math. J., 28, 305–328.
Konyagin, S. V. and Temlyakov, V. N. (2003a), Convergence of greedy approximation I. General systems, Studia Mathematica, 159 (1), 143–160.
Konyagin, S. V. and Temlyakov, V. N. (2003b), Convergence of greedy approximation II. The trigonometric system, Studia Mathematica, 159 (2), 161–184.
Konyagin, S. V. and Temlyakov, V. N. (2004), Some error estimates in learning theory, in D. K., Dimitrov, G., Nikolov and R., Uluchev, eds. Approximation Theory: A Volume Dedicated to Borislav Bojanov (Sofia: Marin Drinov Acad. Publ. House), pp. 126–144.
Konyagin, S. V. and Temlyakov, V. N. (2005), Convergence of greedy approximation for the trigonometric system, Analysis Mathematica, 31, 85–115.
Konyagin, S. V. and Temlyakov, V. N. (2007), The entropy in learning theory. Error estimates, Construct. Approx., 25, 1–27.
Körner, T. W. (1996), Divergence of decreasing rearranged Fourier series, Ann. Math., 144, 167–180.
Körner, T. W. (1999), Decreasing rearranged Fourier series, J. Fourier Anal. Appl., 5, 1–19.
Korobov, N. M. (1959), On the approximate computation of multiple integrals, Dokl. Akad. Nauk SSSR, 124, 1207–1210.
Kuelbs, J. and Li, W. V. (1993), Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal., 116, 133–157.
Kuipers, L. and Niederreiter, H. (1974), Uniform Distribution of Sequences (New York: Wiley).
Lebesgue, H. (1909), Sur les intégrales singuliéres, Ann. Fac. Sci. Univ. Toulouse (3), 1, 25–117.
Lee, W. S., Bartlett, P. L. and Williamson, R. C. (1996), Efficient agnostic learning of neural networks with bounded fan-in, IEEE Trans. Inf. Theory, 42(6), 2118–2132.
Lee, W. S., Bartlett, P. and Williamson, R. (1998), The importance of convexity in learning with square loss, IEEE Trans. Inf. Theory, 44, 1974–1980.
Levenshtein, V. I. (1982), Bounds on the maximal cardinality of a code with bounded modules of the inner product, Sov. Math. Dokl., 25, 526–531.
Levenshtein, V. I. (1983), Bounds for packings of metric spaces and some of their applications, Problemy Kibernetiki, 40, 43–110.
Lifshits, M. A. and Tsirelson, B. S. (1986), Small deviations of Gaussian fields, Teor. Probab. Appl., 31, 557–558.
Lindenstrauss, J. and Tzafriri, L. (1977), Classical Banach Spaces I (Berlin: Springer-Verlag).
Liu, E. and Temlyakov, V. (2010), Orthogonal super greedy algorithm and applications in compressed sensing, IMI Preprint, http://imi.cas.sc.edu/IMI/reports/2010, 10:01, 1–21.
Livshitz, E. D. (2003), Convergence of greedy algorithms in Banach spaces, Math. Notes, 73, 342–368.
Livshitz, E. D. (2006), On the recursive greedy algorithm, Izv.RAN.Ser.Mat., 70, 95–116.
Livshitz, E. D. (2007), Optimality of the greedy algorithm for some function classes, Mat. Sb., 198, 95–114.
Livshitz, E. D. (2009), On lower estimates of rate of convergence of greedy algorithms, Izv. RAN, Ser. Matem., 73, 125–144.
Livshitz, E. D. (2010), On the optimality of Orthogonal Greedy Algorithm for M-coherent dictionaries, Preprint, arXiv:1003.5349v1, 1–14.
Livshitz, E. D. and Temlyakov, V. N. (2001), On the convergence of Weak Greedy Algorithms, Trudy. Mat. Inst. Steklov, 232, 236–247.
Livshitz, E. D. and Temlyakov, V. N. (2003), Two lower estimates in greedy approximation, Construct. Approx., 19, 509–523.
Lugosi, G. (2002), Pattern classification and learning theory, in Principles of Nonparametric Learning (Viena: Springer), pp. 5–62.
Lutoborski, A. and Temlyakov, V. N. (2003), Vector greedy algorithms, J. Complexity, 19, 458–473.
Maiorov, V. E. (1978), On various widths of the class in the space Lq, Izv. Akad.Nauk SSSR Ser. Mat., 42, 773–788; English translation in Math. USSR-Izv. (1979), 13.
Mallat, S. and Zhang, Z. (1993), Matching pursuit in a time-frequency dictionary, IEEE Trans. Signal Proc., 41, 3397–3415.
Matoušsek, J. (1999), Geometric Discrepancy (New York: Springer-Verlag).
Mendelson, S. (2003), A few notes on statistical learning theory, in Advanced Lectures in Machine Learning, LNCS, 2600 (Berlin: Springer), pp. 1–40.
Needell, D. and Vershynin, R. (2009), Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit, Found. Comp. Math., 9, 317–334.
Nelson, J. L. and Temlyakov, V. N. (2008), On the size of incoherent systems, Preprint http://dsp.rice.edu/cs, 1–14.
Niederreiter, H., Tichy, R. F. and Turnwald, G. (1990), An inequality for differences of distribution functionsArch. Math., 54, 166–172.
Nielsen, M. (2009), Trigonometric quasi-greedy bases for Lp (T; w), Rocky Mountain J. Math., 39, 1267–1278.
Nikol'skii, S. N. (1975), Approximation of Functions of Several Variables and Embedding Theorems (Berlin: Springer-Verlag).
Novak, E. (1988), Deterministic and Stochastic Error Bounds in Numerical Analysis, Lecture Notes in Mathematics 1349 (Berlin: Springer-Verlag).
Novak, E. and Wozniakowski, H. (2001), When are integration and discrepancy tractable?, FoCM Proc., London Math. Soc. Lecture Notes Series, 284, 211–266.
Oswald, P. (2001), Greedy algorithms and best m-term approximation with respect to biorthogonal systems, J. Fourier Anal. Appl., 7, 325–341.
Pajor, A. and Tomczak-Yaegermann, N. (1986), Subspaces of small codimension of finite-dimensional Banach spaces, Proc. Amer. Math. Soc., 97, 637–642.
Petrushev, P. (1988), Direct and Converse Theorems for Spline and Rational Approximation and Besov Spaces, Lecture Notes in Mathematics 1302 (Berlin: Springer-Verlag), pp. 363–377.
Pisier, G. (1989), The Volume of Convex Bodies and Banach Space Geometry (Cambridge: Cambridge University Press).
Poggio, T. and Smale, S. (2003), The mathematics of learning: dealing with data, Not. Amer. Math. Soc., 50, 537–544.
Pollard, D. (1984), Convergence of Stochastic Processes (New York: Springer-Verlag).
Roth, K. F. (1954), On irregularities of distribution, Mathematica, 1, 73–79.
Roth, K. F. (1976), On irregularities of distribution. II, Commun. Pure Appl. Math., 29, 749–754.
Roth, K. F. (1979), On irregularities of distribution. III, Acta Arith., 35, 373–384.
Roth, K. F. (1980), On irregularities of distribution. IV, Acta Arith., 37, 67–75.
Schmidt, E. (1906), Zur Theorie der linearen und nichtlinearen Integralgleichungen. I, Math. Annalen, 63, 433–476.
Schmidt, W. M. (1972), Irregularities of distribution. VII, Acta Arith., 21, 45–50.
Schmidt, W. M. (1977), Irregularities of distribution. X, in Number Theory and Algebra (New York: Academic Press), pp. 311–329.
Schütt, C. (1984), Entropy numbers of diagonal operators between symmetric Banach spaces, J. Approx. Theory, 40, 121–128.
Sil'nichenko, A. V. (2004), Rate of convergence of greedy algorithms, Mat. Zametki, 76, 628–632.
Skriganov, M. M. (1994), Constructions of uniform distributions in terms of geometry of numbers, Algebra Anal., 6, 200–230.
Smolyak, S. A. (1960), The ∈-entropy of the classes and, in the metric L2, Dokl. Akad. Nauk SSSR, 131, 30–33.
Sobolev, S. L. (1974), Introduction to the Theory of Cubature Formulas (Moscow: Nauka).
Stromberg, T. and Heath, R. Jr. (2003), Grassmannian frames with applications to coding and communications, Appl. Comput. Harm. Anal., 14, 257–275.
Sudakov, V. N. (1971), Gaussian random processes and measures of solid angles in Hilbert spaces, Sov. Math. Dokl., 12, 412–415.
Talagrand, M. (1994), The small ball problem for the Brownian sheet, Ann. Prob., 22, 1331–1354.
Talagrand, M. (2005), The Generic Chaining (Berlin: Springer).
Temlyakov, V. N. (1988a), Approximation by elements of a finite dimensional subspace of functions from various Sobolev or Nikol'skii spaces, Matem. Zametki, 43, 770–786; English translation in Math. Notes, 43, 444–454.
Temlyakov, V. N. (1988b), On estimates of ∈-entropy and widths of classes of functions with bounded mixed derivative or difference, Dokl. Akad. Nauk SSSR, 301, 288–291; English translation in Sov. Math. Dokl., 38, 84–87.
Temlyakov, V. N. (1989a), Approximation of functions with bounded mixed derivative, Proc. Steklov Institute, 1.
Temlyakov, V. N. (1989b), Estimates of the asymptotic characteristics of classes of functions with bounded mixed derivative or difference, Trudy Matem. Inst. Steklov, 189, 138–168; English translation in Proc. Steklov Inst. Math. (1990), 4, 161–197.
Temlyakov, V. N. (1990), On a way of obtaining lower estimates for the errors of quadrature formulas, Matem. Sbornik, 181, 1403–1413; English translation in Math. USSR Sbornik, 71.
Temlyakov, V. N. (1993a), Approximation of Periodic Functions (New York: Nova Science Publishers, Inc.).
Temlyakov, V. N. (1993b), Bilinear approximation and related questions, Proc. Steklov Inst. Math., 4, 245–265.
Temlyakov, V. N. (1995a), An inequality for trigonometric polynomials and its application for estimating the entropy numbers, J. Complexity, 11, 293–307.
Temlyakov, V. N. (1995b), Some inequalities for multivariate Haar polynomials, East J. Approx., 1, 61–72.
Temlyakov, V. N. (1998a), The best m-term approximation and greedy algorithms, Adv. Comp. Math., 8, 249–265.
Temlyakov, V. N. (1998b), Nonlinear m-term approximation with regard to the multivariate Haar system, East J. Approx., 4, 87–106.
Temlyakov, V. N. (1998c), Greedy algorithm and m-term trigonometric approximation, Construct. Approx., 14, 569–587.
Temlyakov, V. N. (1998d), Nonlinear Kolmogorov's widths, Matem. Zametki, 63, 891–902.
Temlyakov, V. N. (1998e), On two problems in the multivariate approximation, East J. Approx., 4, 505–514.
Temlyakov, V. N. (1999), Greedy algorithms and m-term approximation with regard to redundant dictionaries, J. Approx. Theory, 98, 117–145.
Temlyakov, V. N. (2000a), Greedy algorithms with regard to multivariate systems with special structure, Construct. Approx., 16, 399–425.
Temlyakov, V. N. (2000b), Weak greedy algorithms, Adv. Comp. Math., 12, 213–227.
Temlyakov, V. N. (2001a), Lecture notes on approximation theory, University of South Carolina, Chapter I, pp. 1–20.
Temlyakov, V. N. (2001b), Greedy algorithms in Banach spaces, Adv. Comp. Math., 14, 277–292.
Temlyakov, V. N. (2002a), Universal bases and greedy algorithms for anisotropic function classes, Construct. Approx., 18, 529–550.
Temlyakov, V. N. (2002b), A criterion for convergence of Weak Greedy Algorithms, Adv. Comput. Math., 17, 269–280.
Temlyakov, V. N. (2002c), Nonlinear approximation with regard to bases, in C. K., Chui, L., Schumaker and J., Stöckler, eds., Approximation Theory X (Nashville, TN: Vanderbilt University Press), pp. 373–402.
Temlyakov, V. N. (2003a), Nonlinear methods of approximation, Found. Comput. Math., 3, 33–107.
Temlyakov, V. N. (2003b), Cubature formulas, discrepancy, and nonlinear approximation, J. Complexity, 19, 352–391.
Temlyakov, V. N. (2005a), Greedy type algorithms in Banach spaces and applications, Construct. Approx., 21, 257–292.
Temlyakov, V. N. (2005b), Greedy algorithms with restricted depth search, Proc. Steklov Inst. Math., 248, 255–267.
Temlyakov, V. N. (2006a), Greedy approximations, in Foundations of Computational Mathematics, Santander 2005, London Mathematical Society Lecture Notes Series, 331 (Cambridge: Cambridge University Press), pp. 371–394.
Temlyakov, V. N. (2006b), Greedy approximations with regard to bases, in Proceedings of the International Congress of Mathematicians, Vol.II (Zurich: European Mathematical Society), pp. 1479–1504.
Temlyakov, V. N. (2006c), Relaxation in greedy approximation, IMI-Preprint, 03, 1–26; http://imi.cas.sc.edu/IMI/reports/2006/reports/0603.pdf
Temlyakov, V. N. (2006d), Optimal estimators in learning theory, in T., Figiel and A., Kamont, eds., Approximation and Probability, Banach Center Publications 72 (Warsaw: Warsaw University of Technology), pp. 341–366.
Temlyakov, V. N. (2006e), On universal estimators in learning theory, Proc. Steklov Inst. Math., 255, 244–259.
Temlyakov, V. N. (2007a), Greedy expansions in Banach spaces, Adv. Comput. Math., 26, 431-449.
Temlyakov, V. N. (2007b), Greedy algorithms with prescribed coefficients, J. Fourier Anal. Appl., 71–86.
Temlyakov, V. N. (2008a), Approximation in learning theory, Construct. Approx., 27, 33–74.
Temlyakov, V. N. (2008b), Greedy approximation, Acta Numerica, 17, 235–409.
Temlyakov, V. N. (2008c), Relaxation in greedy approximation, Construct. Approx., 28, 1–25.
Temlyakov, V. N. and Zheltov, P. (2010), On performance of greedy algorithms, IMI-Preprint, 10:02, 1–13; http://imi.cas.sc.edu/IMI/reports/2010/reports/1002.pdf
Tropp, J. A. (2004), Greed is good: algorithmic results for sparse approximation, IEEE Trans. Inform. Theory, 50, 2231–2242.
Tropp, J. A. and Gilbert, A. C. (2007), Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inform. Theory, 52, 4655–4666.
Van de Geer, S. (2000), Empirical Process in M-Estimaton (New York: Cambridge University Press).
Vapnik, V. (1998), Statistical Learning Theory (New York: John Wiley & Sons, Inc.).
Vilenkin, I. V. (1967), Plane nets of integration, Zhur. Vychisl. Mat. i Mat. Fis., 7, 189–196; English translation in USSR Comp. Math. Math. Phys., 7, 258–267.
Wojtaszczyk, P. (1997), On unconditional polynomial bases in Lp and Bergman spaces, Construct. Approx., 13, 1–15.
Wojtaszczyk, P. (2000), Greedy algorithms for general systems, J. Approx. Theory, 107, 293–314.
Wojtaszczyk, P. (2002a), Greedy type bases in Banach spaces, Construct. Funct. Theory (Sofia: DARBA), pp. 1–20.
Wojtaszczyk, P. (2002b), Existence of best m-term approximation, Functiones et Approximatio, XXX, 127–133.
Wojtaszczyk, P. (2006), Greediness of the Haar system in rearrangement invariant spaces, in T., Figiel and A., Kamont, eds., Approximation and Probability, Banach Center Publications 72 (Warsaw: Warsaw University of Technology), pp. 385–395.
Yang, Y. and Barron, A. (1999), Information-theoretic determination of minimax rates of convergence, Ann. Stat., 27, 1564-1599.
Zygmund, A. (1959), Trigonometric Series (Cambridge: Cambridge University Press).

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