The authors study the tractability and strong tractability of a multivariate integration problem in the worst case setting for weighted 1-periodic continuous functions spaces of d coordinates with absolutely convergen...The authors study the tractability and strong tractability of a multivariate integration problem in the worst case setting for weighted 1-periodic continuous functions spaces of d coordinates with absolutely convergent Fourier series. The authors reduce the initial error by a factor ε for functions from the unit ball of the weighted periodic continuous functions spaces. Tractability is the minimal number of function samples required to solve the problem in polynomial in ε^-1 and d, and the strong tractability is the presence of only a polynomial dependence in ε^-1. This problem has been recently studied for quasi-Monte Carlo quadrature rules, quadrature rules with non-negative coefficients, and rules for which all quadrature weights are arbitrary for weighted Korobov spaces of smooth periodic functions of d variables. The authors show that the tractability and strong tractability of a multivariate integration problem in worst case setting hold for the weighted periodic continuous functions spaces with absolutely convergent Fourier series under the same assumptions as in Ref,[14] on the weights of the Korobov space for quasi-Monte Carlo rules and rules for which all quadrature weights are non-negative. The arguments are not constructive.展开更多
The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on which is the space of polynomial splines with irregularly distributed nodes T = where is a real sequence such that constitutes a Rie...The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on which is the space of polynomial splines with irregularly distributed nodes T = where is a real sequence such that constitutes a Riesz basis for L2([ -π, π]) . From these results, the asymptotic relationis proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley- Wiener class.展开更多
基金Project supported by the National Natural Science Foundation of China(10671019)Research Fund for the Doctoral Program Higher Education(20050027007)Beijing Educational Committee(2002Kj112)
文摘The authors study the tractability and strong tractability of a multivariate integration problem in the worst case setting for weighted 1-periodic continuous functions spaces of d coordinates with absolutely convergent Fourier series. The authors reduce the initial error by a factor ε for functions from the unit ball of the weighted periodic continuous functions spaces. Tractability is the minimal number of function samples required to solve the problem in polynomial in ε^-1 and d, and the strong tractability is the presence of only a polynomial dependence in ε^-1. This problem has been recently studied for quasi-Monte Carlo quadrature rules, quadrature rules with non-negative coefficients, and rules for which all quadrature weights are arbitrary for weighted Korobov spaces of smooth periodic functions of d variables. The authors show that the tractability and strong tractability of a multivariate integration problem in worst case setting hold for the weighted periodic continuous functions spaces with absolutely convergent Fourier series under the same assumptions as in Ref,[14] on the weights of the Korobov space for quasi-Monte Carlo rules and rules for which all quadrature weights are non-negative. The arguments are not constructive.
文摘The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on which is the space of polynomial splines with irregularly distributed nodes T = where is a real sequence such that constitutes a Riesz basis for L2([ -π, π]) . From these results, the asymptotic relationis proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley- Wiener class.
