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  • Authors: Dinh Dũng, Tino Ullrich (2012)

  • In this paper, we study linear trigonometric hyperbolic cross approximations, Kol-mogorov n-widths dn(W; H ), and "-dimensions n"(W; H ) of periodic d-variate func-tion classes W with anisotropic smoothness, where d may be large. We are interested in nding the accurate dependence of dn(W; H ) and n"(W; H ) as a function of two variables n, d and ", d, respectively. Recall that n, the dimension of the approximat-ing subspace, is the main parameter in the study of convergence rates with respect to n going to in nity. However, the parameter d may seriously affect this rate when d is large. We construct linear approximations of functions from W by trigonomet-ric polynomials with frequencies from hyperbolic crosses and prove upper bounds for the error measured in isotropic Sobolev spaces...

  • Book; Book chapter; Dataset


  • Authors: Dinh Dũng, Tino Ullrich (2015)

  • We prove lower bounds for the error of optimal cubature formulae for d-variate functions from Besov spaces of mixed smoothness in the case , and , where is either the d-dimensional torus or the d-dimensional unit cube . In addition, we prove upper bounds for QMC integration on the Fibonacci-lattice for bivariate periodic functions from in the case , and . A non-periodic modification of this classical formula yields upper bounds for if . In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from and indicate that a corresponding result is most likely also true in case . This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature fo...