Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-29T23:02:06.137Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  05 August 2012

Vladimir 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
Greedy Approximation , pp. 405 - 414
Publisher: Cambridge University Press
Print publication year: 2011

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

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.Google Scholar
Alon, N. (2003), Problems and results in extremal combinatorics, Discrete Math., 273, 31–53.Google Scholar
Baishanski, B. M. (1983), Approximation by polynomials of given length, Illinois J. Math., 27, 449–458.Google Scholar
Bakhvalov, N. S. (1959), On the approximate computation of multiple integrals, Vestnik Moskov. Univ. Ser. Mat. Mekh. Astr. Fiz. Khim., 4, 3–18.Google Scholar
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.Google Scholar
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.Google Scholar
Barron, A. R. (1991), Complexity regularization with applications to artificial neural networks. In Nonparametric Functional Estimation, G., Roussas, ed. (Dordrecht: Kluwer), pp. 561–576.CrossRef
Barron, A. R. (1993), Universal approximation bounds for superposition of n sigmoidal functions, IEEE Trans. Inf. Theory, 39, 930–945.Google Scholar
Barron, A., Birgé, L. and Massart, P. (1999), Risk bounds for model selection via penalization, Prob. Theory Related Fields, 113, 301–413.CrossRefGoogle Scholar
Barron, A., Cohen, A., Dahmen, W. and Devore, R. (2008), Approximation and learning by greedy algorithms, Ann. Stat., 36, 64–94.Google Scholar
Bass, R. F. (1988), Probability estimates for multiparameter Brownian processes, Ann. Prob., 16, 251–264.Google Scholar
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.Google Scholar
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.Google Scholar
Bilyk, D. and Lacey, M. (2008), On the small ball inequality in three dimensions, Duke Math J., 143, 81–115.Google Scholar
Bilyk, D., Lacey, M. and Vagharshakyan, A. (2008), On the small ball inequality in all dimensions, J. Func. Anal., 254, 2470–2502.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Candes, E. (2008), The restricted isometry property and its implications for compressed sensing, C. R. Acad. Sci. Paris, Ser. I 346, 589–592.Google Scholar
Candes, E., Romberg, J. and Tao, T. (2006), Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 59, 1207–1223.Google Scholar
Candes, E. and Tao, T. (2005), Decoding by linear programming, IEEE Trans. Inform. Theory, 51, 4203–4215.Google Scholar
Carl, B. (1981), Entropy numbers, s-numbers, and eigenvalue problems, J. Func. Anal., 41, 290–306.Google Scholar
Carlitz, L. and Uchiyama, S. (1957), Bounds for exponential sums, Duke Math. J., 24, 37–41.Google Scholar
Chazelle, B. (2000), The Discrepancy Method, (Cambridge: Cambridge University Press).
Chen, W. W. L. (1980), On irregularities of distribution, Mathematika, 27, 153–170.Google Scholar
Chen, S. S., Donoho, D. L. and Saunders, M. A. (2001), Atomic decomposition by basis pursuit, SIAM Rev., 43, 129–159.Google Scholar
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.Google Scholar
Cohen, A., Devore, R. A. and Hochmuth, R. (2000), Restricted nonlinear approximation, Construct. Approx., 16, 85–113.Google Scholar
Coifman, R. R. and Wickerhauser, M. V. (1992), Entropy-based algorithms for best-basis selection, IEEE Trans. Inform. Theory, 38, 713–718.Google Scholar
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.Google Scholar
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.Google Scholar
Van Der Corput, J. G. (1935a), Verteilungsfunktionen. I, Proc. Kon. Ned. Akad.v. Wetensch., 38, 813–821.Google Scholar
Van Der Corput, J. G. (1935b), Verteilungsfunktionen. II, Proc. Kon. Ned. Akad.v. Wetensch., 38, 1058–1066.Google Scholar
Cucker, F. and Smale, S. (2001), On the mathematical foundations of learning, Bull. AMS, 39, 1–49.CrossRefGoogle Scholar
Dai, W. and Milenkovich, O. (2009), Subspace pursuit for compressive sensing signal reconstruction, IEEE Trans. Inform. Theory, 55, 2230–2249.Google Scholar
Davenport, H. (1956), Note on irregularities of distribution, Mathematika, 3, 131–135.Google Scholar
Davis, G., Mallat, S. and Avellaneda, M. (1997), Adaptive greedy approximations, Construct. Approx., 13, 57–98.Google Scholar
Devore, R. A. (1998), Nonlinear approximation, Acta Numerica, 7, 51–150.Google Scholar
Devore, R. A. (2006), Optimal computation, ICM Proc., Madrid, 1, 187–215.Google Scholar
Devore, R. A. (2007), Deterministic constructions of compressed sensing matricesJ. Complex., 23, 918–925.Google Scholar
Devore, R. A., Jawerth, B. and Popov, V. (1992), Compression of wavelet decompositions, Amer. J. Math., 114, 737–785.Google Scholar
Devore, R. A., Konyagin, S. V. and Temlyakov, V. N. (1998), Hyperbolic wavelet approximation, Construct. Approx., 14, 1–26.Google Scholar
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.Google Scholar
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.Google Scholar
Devore, R. A. and Temlyakov, V. N. (1996), Some remarks on Greedy Algorithms, Adv. Comp. Math., 5, 173–187.Google Scholar
Devore, R. A. and Temlyakov, V. N. (1997), Nonlinear approximation in finite-dimensional spaces, J. Complexity, 13, 489–508.CrossRefGoogle Scholar
Devore, R. A., Kerkyacharian, G., Picard, D. and Temlyakov, V. (2004), On mathematical methods of learning, IMI Preprints, 10, 1–24.Google Scholar
Devore, R. A., Kerkyacharian, G., Picard, D. and Temlyakov, V. (2006), Mathematical methods for supervised learning, Found. Comput. Math., 6, 3–58.Google Scholar
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.Google Scholar
Dilworth, S. J., Kutzarova, D. and Temlyakov, V. (2002), Convergence of some Greedy Algorithms in Banach spaces, J. Fourier Anal. Applic., 8, 489–505.Google Scholar
Dilworth, S. J., Kutzarova, D. and Wojtaszczyk, P. (2002), On approximate ℓ1 systems in Banach spaces, J. Approx. Theory, 114, 214–241.Google Scholar
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.Google Scholar
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.Google Scholar
Donoho, D. L. (1993), Unconditional bases are optimal bases for data compression and for statistical estimation, Appl. Comput. Harmon. Anal., 1, 100–115.Google Scholar
Donoho, D. L. (1997), CART and Best-Ortho-Basis: a connection, Ann. Stat., 25, 1870–1911.Google Scholar
Donoho, D. L. (2001), Sparse components of images and optimal atomic decompositions, Construct. Approx., 17, 353–382.Google Scholar
Donoho, D. L. (2006), Compressed sensing, IEEE Trans. Inform. Theory, 52, 1289–1306.Google Scholar
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.Google Scholar
Donoho, D. L., Elad, M. and Temlyakov, V. N. (2007), On the Lebesgue type inequalities for greedy approximation, J. Approx. Theory, 147, 185–195.Google Scholar
Donoho, D. L. and Johnstone, I. (1994), Ideal spatial adaptation via wavelet shrinkage, Biometrica, 81, 425–455.Google Scholar
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.Google Scholar
Fefferman, C. and Stein, E. (1972), Hp spaces of several variables, Acta Math., 129, 137–193.Google Scholar
Figiel, T., Johnson, W. B. and Schechtman, G. (1988), Factorization of natural embeddings of into Lr, I, Studia Mathematica, 89, 79–103.Google Scholar
Frazier, M. and Jawerth, B. (1990), A discrete transform and decomposition of distribution spaces, J. Funct. Anal., 93, 34–170.Google Scholar
Friedman, J. H. and Stuetzle, W. (1981), Projection pursuit regression, J. Amer. Stat. Assoc., 76, 817–823.Google Scholar
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.Google Scholar
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.Google Scholar
Galatenko, V. V. and Livshitz, E. D. (2003), On convergence of approximate weak greedy algorithms, East J. Approx., 9, 43–49.Google Scholar
Galatenko, V. V. and Livshitz, E. D. (2005), Generalized approximate weak greedy algorithms, Math. Notes, 78, 170–184.CrossRefGoogle Scholar
Ganichev, M. and Kalton, N. J. (2003), Convergence of the Weak Dual Greedy Algorithm in Lp-spaces, J. Approx. Theory, 124, 89–95.Google Scholar
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.Google Scholar
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.Google Scholar
Gluskin, E. D. (1986), An octahedron is poorly approximated by random subspaces, Functsional. Anal. i Prilozhen., 20, 14–20.Google Scholar
Gogyan, S. (2005), Greedy algorithm with regard to Haar subsystems, East J. Approx., 11, 221–236.Google Scholar
Gogyan, S. (2009), On convergence of Weak Thresholding Greedy Algorithm in L1(0, 1), J. Approx. Theory, 161, 49–64.Google Scholar
Gribonval, R. and Nielsen, M. (2001a), Approximate Weak Greedy Algorithms, Adv. Comput. Math., 14, 361–368.Google Scholar
Gribonval, R. and Nielsen, M. (2001b), Some remarks on non-linear approximation with Schauder bases, East J. Approx. 7, 267–285.Google Scholar
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.Google Scholar
Halton, J. H. and Zaremba, S. K. (1969), The extreme and L2 discrepancies of some plane sets, Monats. für Math., 73, 316–328.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, 279–302.Google Scholar
Hitczenko, P. and Kwapien, S. (1994), On Rademacher series, Prog. Prob., 35, 31–36.Google Scholar
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.Google Scholar
Jones, L. (1987), On a conjecture of Huber concerning the convergence of projection pursuit regression, Ann. Stat., 15, 880–882.Google Scholar
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.Google Scholar
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.Google Scholar
Kashin, B. S. (1975), On widths of octahedrons, Uspekhi Matem. Nauk, 30, 251–252.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Kashin, B. S. and Temlyakov, V. N. (2003), The volume estimates and their applications, East J. Approx., 9, 469–485.Google Scholar
Kashin, B. S. and Temlyakov, V. N. (2007), A remark on compressed sensing, Math. Notes, 82, 748–755.CrossRefGoogle Scholar
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.Google Scholar
Kerkyacharian, G. and Picard, D. (2004), Entropy, universal coding, approximation, and bases properties, Construct. Approx., 20, 1–37.Google Scholar
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.Google Scholar
Konyagin, S. V. and Skopina, M. A. (2001), Comparison of the L1-norms of total and truncated exponential sums, Mat. Zametki, 69, 699–707.Google Scholar
Konyagin, S. V. and Temlyakov, V. N. (1999a), A remark on greedy approximation in Banach spaces, East J. Approx., 5, 365–379.Google Scholar
Konyagin, S. V. and Temlyakov, V. N. (1999b), Rate of convergence of Pure Greedy Algorithm, East J. Approx., 5, 493–499.Google Scholar
Konyagin, S. V. and Temlyakov, V. N. (2002), Greedy approximation with regard to bases and general minimal systems, Serdica Math. J., 28, 305–328.Google Scholar
Konyagin, S. V. and Temlyakov, V. N. (2003a), Convergence of greedy approximation I. General systems, Studia Mathematica, 159 (1), 143–160.Google Scholar
Konyagin, S. V. and Temlyakov, V. N. (2003b), Convergence of greedy approximation II. The trigonometric system, Studia Mathematica, 159 (2), 161–184.Google Scholar
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.Google Scholar
Konyagin, S. V. and Temlyakov, V. N. (2007), The entropy in learning theory. Error estimates, Construct. Approx., 25, 1–27.Google Scholar
Körner, T. W. (1996), Divergence of decreasing rearranged Fourier series, Ann. Math., 144, 167–180.Google Scholar
Körner, T. W. (1999), Decreasing rearranged Fourier series, J. Fourier Anal. Appl., 5, 1–19.Google Scholar
Korobov, N. M. (1959), On the approximate computation of multiple integrals, Dokl. Akad. Nauk SSSR, 124, 1207–1210.Google Scholar
Kuelbs, J. and Li, W. V. (1993), Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal., 116, 133–157.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Levenshtein, V. I. (1983), Bounds for packings of metric spaces and some of their applications, Problemy Kibernetiki, 40, 43–110.Google Scholar
Lifshits, M. A. and Tsirelson, B. S. (1986), Small deviations of Gaussian fields, Teor. Probab. Appl., 31, 557–558.Google Scholar
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.CrossRefGoogle Scholar
Livshitz, E. D. (2006), On the recursive greedy algorithm, Izv.RAN.Ser.Mat., 70, 95–116.Google Scholar
Livshitz, E. D. (2007), Optimality of the greedy algorithm for some function classes, Mat. Sb., 198, 95–114.Google Scholar
Livshitz, E. D. (2009), On lower estimates of rate of convergence of greedy algorithms, Izv. RAN, Ser. Matem., 73, 125–144.Google Scholar
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.Google Scholar
Livshitz, E. D. and Temlyakov, V. N. (2003), Two lower estimates in greedy approximation, Construct. Approx., 19, 509–523.Google Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Mallat, S. and Zhang, Z. (1993), Matching pursuit in a time-frequency dictionary, IEEE Trans. Signal Proc., 41, 3397–3415.Google Scholar
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.Google Scholar
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.Google Scholar
Nielsen, M. (2009), Trigonometric quasi-greedy bases for Lp (T; w), Rocky Mountain J. Math., 39, 1267–1278.Google Scholar
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.Google Scholar
Oswald, P. (2001), Greedy algorithms and best m-term approximation with respect to biorthogonal systems, J. Fourier Anal. Appl., 7, 325–341.Google Scholar
Pajor, A. and Tomczak-Yaegermann, N. (1986), Subspaces of small codimension of finite-dimensional Banach spaces, Proc. Amer. Math. Soc., 97, 637–642.Google Scholar
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.Google Scholar
Pollard, D. (1984), Convergence of Stochastic Processes (New York: Springer-Verlag).
Roth, K. F. (1954), On irregularities of distribution, Mathematica, 1, 73–79.Google Scholar
Roth, K. F. (1976), On irregularities of distribution. II, Commun. Pure Appl. Math., 29, 749–754.Google Scholar
Roth, K. F. (1979), On irregularities of distribution. III, Acta Arith., 35, 373–384.Google Scholar
Roth, K. F. (1980), On irregularities of distribution. IV, Acta Arith., 37, 67–75.Google Scholar
Schmidt, E. (1906), Zur Theorie der linearen und nichtlinearen Integralgleichungen. I, Math. Annalen, 63, 433–476.CrossRefGoogle Scholar
Schmidt, W. M. (1972), Irregularities of distribution. VII, Acta Arith., 21, 45–50.Google Scholar
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.Google Scholar
Sil'nichenko, A. V. (2004), Rate of convergence of greedy algorithms, Mat. Zametki, 76, 628–632.Google Scholar
Skriganov, M. M. (1994), Constructions of uniform distributions in terms of geometry of numbers, Algebra Anal., 6, 200–230.Google Scholar
Smolyak, S. A. (1960), The ∈-entropy of the classes and, in the metric L2, Dokl. Akad. Nauk SSSR, 131, 30–33.Google Scholar
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.Google Scholar
Sudakov, V. N. (1971), Gaussian random processes and measures of solid angles in Hilbert spaces, Sov. Math. Dokl., 12, 412–415.Google Scholar
Talagrand, M. (1994), The small ball problem for the Brownian sheet, Ann. Prob., 22, 1331–1354.Google Scholar
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.Google Scholar
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.Google Scholar
Temlyakov, V. N. (1989a), Approximation of functions with bounded mixed derivative, Proc. Steklov Institute, 1.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Temlyakov, V. N. (1995a), An inequality for trigonometric polynomials and its application for estimating the entropy numbers, J. Complexity, 11, 293–307.Google Scholar
Temlyakov, V. N. (1995b), Some inequalities for multivariate Haar polynomials, East J. Approx., 1, 61–72.Google Scholar
Temlyakov, V. N. (1998a), The best m-term approximation and greedy algorithms, Adv. Comp. Math., 8, 249–265.Google Scholar
Temlyakov, V. N. (1998b), Nonlinear m-term approximation with regard to the multivariate Haar system, East J. Approx., 4, 87–106.Google Scholar
Temlyakov, V. N. (1998c), Greedy algorithm and m-term trigonometric approximation, Construct. Approx., 14, 569–587.Google Scholar
Temlyakov, V. N. (1998d), Nonlinear Kolmogorov's widths, Matem. Zametki, 63, 891–902.Google Scholar
Temlyakov, V. N. (1998e), On two problems in the multivariate approximation, East J. Approx., 4, 505–514.Google Scholar
Temlyakov, V. N. (1999), Greedy algorithms and m-term approximation with regard to redundant dictionaries, J. Approx. Theory, 98, 117–145.Google Scholar
Temlyakov, V. N. (2000a), Greedy algorithms with regard to multivariate systems with special structure, Construct. Approx., 16, 399–425.Google Scholar
Temlyakov, V. N. (2000b), Weak greedy algorithms, Adv. Comp. Math., 12, 213–227.Google Scholar
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.Google Scholar
Temlyakov, V. N. (2002a), Universal bases and greedy algorithms for anisotropic function classes, Construct. Approx., 18, 529–550.Google Scholar
Temlyakov, V. N. (2002b), A criterion for convergence of Weak Greedy Algorithms, Adv. Comput. Math., 17, 269–280.Google Scholar
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.Google Scholar
Temlyakov, V. N. (2003b), Cubature formulas, discrepancy, and nonlinear approximation, J. Complexity, 19, 352–391.Google Scholar
Temlyakov, V. N. (2005a), Greedy type algorithms in Banach spaces and applications, Construct. Approx., 21, 257–292.Google Scholar
Temlyakov, V. N. (2005b), Greedy algorithms with restricted depth search, Proc. Steklov Inst. Math., 248, 255–267.Google Scholar
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.pdfGoogle Scholar
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.Google Scholar
Temlyakov, V. N. (2007a), Greedy expansions in Banach spaces, Adv. Comput. Math., 26, 431-449.Google Scholar
Temlyakov, V. N. (2007b), Greedy algorithms with prescribed coefficients, J. Fourier Anal. Appl., 71–86.Google Scholar
Temlyakov, V. N. (2008a), Approximation in learning theory, Construct. Approx., 27, 33–74.Google Scholar
Temlyakov, V. N. (2008b), Greedy approximation, Acta Numerica, 17, 235–409.Google Scholar
Temlyakov, V. N. (2008c), Relaxation in greedy approximation, Construct. Approx., 28, 1–25.Google Scholar
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.Google Scholar
Tropp, J. A. and Gilbert, A. C. (2007), Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inform. Theory, 52, 4655–4666.Google Scholar
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.Google Scholar
Wojtaszczyk, P. (1997), On unconditional polynomial bases in Lp and Bergman spaces, Construct. Approx., 13, 1–15.Google Scholar
Wojtaszczyk, P. (2000), Greedy algorithms for general systems, J. Approx. Theory, 107, 293–314.Google Scholar
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.Google Scholar
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.CrossRef
Yang, Y. and Barron, A. (1999), Information-theoretic determination of minimax rates of convergence, Ann. Stat., 27, 1564-1599.Google Scholar
Zygmund, A. (1959), Trigonometric Series (Cambridge: Cambridge University Press).

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org 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
  • Vladimir Temlyakov, University of South Carolina
  • Book: Greedy Approximation
  • Online publication: 05 August 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511762291.008
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
  • Vladimir Temlyakov, University of South Carolina
  • Book: Greedy Approximation
  • Online publication: 05 August 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511762291.008
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
  • Vladimir Temlyakov, University of South Carolina
  • Book: Greedy Approximation
  • Online publication: 05 August 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511762291.008
Available formats
×