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Integration and approximation of univariate functions

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Average-Case Analysis of Numerical Problems

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1.4. Notes and References.

  • Sacks, J., and Ylvisaker, D. (1966), Designs for regression with correlated errors, Ann. Math. Statist. 37, 68–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1968), Designs for regression problems with correlated errors; many parameters, Ann. Math. Statist. 39, 49–69.

    Article  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1970a), Design for regression problems with correlated errors III, Ann. Math. Statist. 41, 2057–2074.

    Article  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1970b), Statistical design and integral approximation, in: Proc. 12th Bienn. Semin. Can. Math. Congr., R. Pyke, ed., pp. 115–136, Can. Math. Soc., Montreal.

    Google Scholar 

  • Benhenni, K., and Cambanis, S. (1992a), Sampling designs for estimating integrals of stochastic processes, Ann. Statist. 20, 161–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Benhenni, K., and Cambanis, S. (1992b), Sampling designs for estimating integrals of stochastic processes using quadratic mean derivatives, in: Approximation theory, G. A. Anastassiou, ed., pp. 93–123, Dekker, New York.

    Google Scholar 

  • Su, Y., and Cambanis, S. (1993), Sampling designs for estimation of a random process, Stochastic Processes Appl. 46, 47–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Istas, J., and Laredo, C. (1994), Estimation d'intégrales de processus aléatoires à partir d'observations discrétisées, C. R. Acad. Sci. 319, 85–88.

    MathSciNet  Google Scholar 

  • Istas, J., and Laredo, C. (1997), Estimating functionals of a stochastic process, Adv. Appl. Prob. 29, 249–270.

    Article  MathSciNet  MATH  Google Scholar 

  • Müller-Gronbach, T. (1996), Optimal designs for approximating the path of a stochastic process, J. Statist. Planning Inf. 49, 371–385.

    Article  MathSciNet  MATH  Google Scholar 

  • Novak, E., Ritter, K., and Woźniakowski, H. (1995), Average case optimality of a hybrid secant-bisection method, Math. Comp. 64, 1517–1539.

    Article  MathSciNet  MATH  Google Scholar 

  • Ritter, K. (1996a), Asymptotic optimality of regular sequence designs, Ann. Statist. 24, 2081–2096.

    Article  MathSciNet  MATH  Google Scholar 

  • Müller-Gronbach, T. (1996), Optimal designs for approximating the path of a stochastic process, J. Statist. Planning Inf. 49, 371–385.

    Article  MathSciNet  MATH  Google Scholar 

  • Novak, E., Ritter, K., and Woźniakowski, H. (1995), Average case optimality of a hybrid secant-bisection method, Math. Comp. 64, 1517–1539.

    Article  MathSciNet  MATH  Google Scholar 

  • Ritter, K. (1996a), Asymptotic optimality of regular sequence designs, Ann. Statist. 24, 2081–2096.

    Article  MathSciNet  MATH  Google Scholar 

  • Novak, E., Ritter, K., and Woźniakowski, H. (1995), Average case optimality of a hybrid secant-bisection method, Math. Comp. 64, 1517–1539.

    Article  MathSciNet  MATH  Google Scholar 

2.4. Notes and References

  • Barrow, D. L., and Smith, P. W. (1979), Asymptotic properties of optimal quadrature formulas, in: Numerische Integration, G. Hämmerlin, ed., ISNM 45, pp. 54–66, Birkhäuser Verlag, Basel.

    Chapter  Google Scholar 

  • Eubank, R. L., Smith, P. L., and Smith, P. W. (1981), Uniqueness and eventual uniqueness of optimal designs in some time series models, Ann. Statist. 9, 486–493.

    Article  MathSciNet  MATH  Google Scholar 

  • Eubank, R. L., Smith, P. L., and Smith, P. W. (1982), A note on optimal and asymptotically optimal designs for certain time series models, Ann. Statist. 10, 1295–1301.

    Article  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1966), Designs for regression with correlated errors, Ann. Math. Statist. 37, 68–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1970a), Design for regression problems with correlated errors III, Ann. Math. Statist. 41, 2057–2074.

    Article  MathSciNet  MATH  Google Scholar 

  • Benhenni, K., and Cambanis, S. (1992a), Sampling designs for estimating integrals of stochastic processes, Ann. Statist. 20, 161–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Ritter, K. (1996a), Asymptotic optimality of regular sequence designs, Ann. Statist. 24, 2081–2096.

    Article  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1966), Designs for regression with correlated errors, Ann. Math. Statist. 37, 68–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Sacks, J., and Ylvisaker, D. (1970b), Statistical design and integral approximation, in: Proc. 12th Bienn. Semin. Can. Math. Congr., R. Pyke, ed., pp. 115–136, Can. Math. Soc., Montreal.

    Google Scholar 

  • Barrow, D. L., and Smith, P. W. (1979), Asymptotic properties of optimal quadrature formulas, in: Numerische Integration, G. Hämmerlin, ed., ISNM 45, pp. 54–66, Birkhäuser Verlag, Basel.

    Chapter  Google Scholar 

  • Eubank, R. L., Smith, P. L., and Smith, P. W. (1982), A note on optimal and asymptotically optimal designs for certain time series models, Ann. Statist. 10, 1295–1301.

    Article  MathSciNet  MATH  Google Scholar 

  • Benhenni, K., and Cambanis, S. (1992a), Sampling designs for estimating integrals of stochastic processes, Ann. Statist. 20, 161–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Wahba, G. (1971), On the regression design problem of Sacks and Ylvisaker, Ann. Math. Statist. 42, 1035–1043.

    Article  MathSciNet  MATH  Google Scholar 

  • Wahba, G. (1974), Regression design for some equivalence classes of kernels, Ann. Statist. 2, 925–934.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, J., and Kimeldorf, G. S. (1974), Regression designs in autoregressive stochastic processes, Ann. Statist. 2, 520–527.

    Article  MathSciNet  MATH  Google Scholar 

  • Ritter, K. (1996a), Asymptotic optimality of regular sequence designs, Ann. Statist. 24, 2081–2096.

    Article  MathSciNet  MATH  Google Scholar 

  • Temirgaliev, N. (1988), On an application of infinitely divisible distributions to quadrature problems, Anal. Math. 14, 253–258.

    Article  MathSciNet  MATH  Google Scholar 

  • Voronin, S. M., and Skalyga, V. I. (1984), On quadrature formulas, Soviet Math. Dokl. 29, 616–619.

    MathSciNet  MATH  Google Scholar 

  • Lee, D., and Wasilkowski, G. W. (1986), Approximation of linear functionals on a Banach space with a Gaussian measure, J. Complexity 2, 12–43.

    Article  MathSciNet  MATH  Google Scholar 

  • Gao, F. (1993), On the role of computable error estimates in the analysis of numerical approximation algorithms, in: Proceedings of the Smalefest, M. W. Hirsch, J. E. Marsden, and M. Shub, eds., pp. 387–394, Springer-Verlag, New York.

    Google Scholar 

3.6. Notes and References

  • Speckman, P. (1979), L p approximation of autoregressive Gaussian processes, Tech. Rep., Dept. of Statistics, Univ. of Oregon, Eugene.

    Google Scholar 

  • Su, Y., and Cambanis, S. (1993), Sampling designs for estimation of a random process, Stochastic Processes Appl. 46, 47–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Müller-Gronbach, T. (1996), Optimal designs for approximating the path of a stochastic process, J. Statist. Planning Inf. 49, 371–385.

    Article  MathSciNet  MATH  Google Scholar 

  • Ritter, K. (1996c), Average case analysis of numerical problems, Habilitations-schrift, Mathematisches Institut, Univ. Erlangen-Nürnberg.

    MATH  Google Scholar 

  • Papageorgiou, A., and Wasilkowski, G. W. (1990), On the average complexity of multivariate problems, J. Complexity 6, 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  • Maiorov, V. E. (1992), Widths of spaces endowed with a Gaussian measure, Russian Acad. Sci. Dokl. Math. 45, 305–309.

    MathSciNet  Google Scholar 

  • Sun, Yong-Sheng, and Wang, Chengyong (1994), μ-average n-widths on the Wiener space, J. Complexity 10, 428–436.

    Article  MathSciNet  MATH  Google Scholar 

  • Sun, Yong-Sheng, and Wang, Chengyong (1994), μ-average n-widths on the Wiener space, J. Complexity 10, 428–436.

    Article  MathSciNet  MATH  Google Scholar 

  • Wasilkowski, G. W. (1992), On average complexity of global optimization problems, Math. Programming 57, 313–324.

    Article  MathSciNet  MATH  Google Scholar 

  • Wasilkowski, G. W. (1996), Average case complexity of multivariate integration and function approximation, an overview, J. Complexity 12, 257–272.

    Article  MathSciNet  MATH  Google Scholar 

5.4. Notes and References

  • Buslaev, A. P., and Seleznjev, O. V. (1999), On certain extremal problems in the theory of approximation of random processes, East J. Approx. Theory 4, 467–481.

    MathSciNet  MATH  Google Scholar 

  • Cambanis, S., and Masry, E. (1994), Wavelet approximation of deterministic and random signals: convergence properties and rates, IEEE Trans. Inform. Theory IT-40, 1013–1029.

    Article  MathSciNet  MATH  Google Scholar 

  • Seleznjev, O. V. (2000), Spline approximation of random processes and design problems, J. Statist. Planning Inf. 84, 249–262.

    Article  MathSciNet  MATH  Google Scholar 

  • Stein, M. L. (1995c), Predicting integrals of stochastic processes, Ann. Appl. Prob. 5, 158–170.

    Article  MathSciNet  MATH  Google Scholar 

  • Benhenni, K. (1997), Approximation d'intégrales de processus stochastiques, C. R. Acad. Sci. 325, 659–663.

    Article  MathSciNet  Google Scholar 

  • Benhenni, K. (1998), Approximating integrals of stochastic processes: extensions, J. Appl. Probab. 35, 843–855.

    Article  MathSciNet  MATH  Google Scholar 

  • Stein, M. L. (1995c), Predicting integrals of stochastic processes, Ann. Appl. Prob. 5, 158–170.

    Article  MathSciNet  MATH  Google Scholar 

  • Sun, Yong-Sheng, and Wang, Chengyong (1995), Average error bounds of best approximation of continuous functions on the Wiener space, J. Complexity 11, 74–104.

    Article  MathSciNet  MATH  Google Scholar 

  • Stein, M. L. (1995c), Predicting integrals of stochastic processes, Ann. Appl. Prob. 5, 158–170.

    Article  MathSciNet  MATH  Google Scholar 

  • Istas, J., and Laredo, C. (1994), Estimation d'intégrales de processus aléatoires à partir d'observations discrétisées, C. R. Acad. Sci. 319, 85–88.

    MathSciNet  Google Scholar 

  • Istas, J., and Laredo, C. (1997), Estimating functionals of a stochastic process, Adv. Appl. Prob. 29, 249–270.

    Article  MathSciNet  MATH  Google Scholar 

  • Seleznjev, O. V. (1991), Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes, J. Appl. Probab. 28, 17–32.

    Article  MathSciNet  MATH  Google Scholar 

  • Seleznjev, O. V. (1996), Large deviations in the piecewise linear approximation of Gaussian processes with stationary increments, Adv. Appl. Prob. 28, 481–499.

    Article  MathSciNet  MATH  Google Scholar 

  • Hüsler, J. (1999), Extremes of Gaussian processes, on results of Piterbarg and Seleznjev, Stat. Prob. Letters 44, 251–258.

    Article  MathSciNet  MATH  Google Scholar 

  • Istas, J., and Laredo, C. (1997), Estimating functionals of a stochastic process, Adv. Appl. Prob. 29, 249–270.

    Article  MathSciNet  MATH  Google Scholar 

  • Stein, M. L. (1995c), Predicting integrals of stochastic processes, Ann. Appl. Prob. 5, 158–170.

    Article  MathSciNet  MATH  Google Scholar 

  • Benhenni, K., and Istas, J. (1998), Minimax results for estimating integrals of analytic processes, ESAIM, Probab. Stat. 2, 109–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Weba, M. (1991a), Quadrature of smooth stochastic processes, Probab. Theory Relat. Fields 87, 333–347.

    Article  MathSciNet  MATH  Google Scholar 

  • Weba, M. (1991b), Interpolation of random functions, Numer. Math. 59, 739–746.

    Article  MathSciNet  MATH  Google Scholar 

  • Weba, M. (1992), Simulation and approximation of stochastic processes by spline functions, SIAM J. Sci. Stat. Comput. 13, 1085–1096.

    Article  MathSciNet  MATH  Google Scholar 

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Klaus Ritter

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Ritter, K. (2000). Integration and approximation of univariate functions. In: Ritter, K. (eds) Average-Case Analysis of Numerical Problems. Lecture Notes in Mathematics, vol 1733. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103938

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