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Abstract In this paper, we investigate optimal linear approximations (n -approximation numbers) of the embeddings from the Sobolev spaces H r (r > 0) for various equivalent norms and the Gevrey type spaces G α , β (α , β > 0) on the sphere S d and on the ball B d , where the approximation error is measured in the L 2 -norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in n and the dimension d. We emphasize that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension d and n. As a consequence we obtain that for the absolute error criterion the approximation problems I d : H r → L 2 are weakly tractable if and only if r > 1 , not uniformly weakly tractable, and do not suffer from the curse of dimensionality. We also prove that for any α , β > 0 , the approximation problems I d : G α , β → L 2 are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if α ≥ 1. [ABSTRACT FROM AUTHOR]

In this paper we investigate multivariate integration in weighted unanchored Sobolev spaces of smoothness of arbitrarily high order. As quadrature points we employ higher order polynomial lattice point sets over F 2 which are randomly digitally shifted and then folded using the tent transformation. We first prove the existence of good higher order polynomial lattice rules which achieve the optimal rate of the mean square worst-case error, while reducing the required degree of modulus by half as compared to higher order polynomial lattice rules whose quadrature points are randomly digitally shifted but not folded using the tent transformation. Thus we are able to restrict the search space of generating vectors significantly. We then study the component-by-component construction as an explicit means of obtaining good higher order polynomial lattice rules. In a way analogous to Baldeaux et al. (2012), we show how to calculate the quality criterion efficiently and how to obtain the fast component-by-component construction using the fast Fourier transform. Our result generalizes the previous result shown by Cristea et al. (2007), in which the degree of smoothness is fixed at 2 and classical polynomial lattice rules are considered. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the asymptotic behavior of the Gelfand, Kolmogorov and Weyl numbers of Sobolev embeddings in weighted function spaces of Besov and Triebel–Lizorkin type with polynomial weights. The exact estimates for the Gelfand and Kolmogorov numbers are obtained in the so-called non-limiting case and complement those previous results in the quasi-Banach setting. Also, the asymptotic order of the Weyl numbers is established by means of the basic Weyl estimates in the finite-dimensional spaces improved recently as well as flexible uses of the operator ideal technique. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the best approximation of functions by spherical polynomials and the approximation by Fourier partial summation operators, Vallée–Poussin operators, Cesàro operators, and Abel operators, on the Sobolev space on the sphere with a Gaussian measure, and obtain the average error estimates. We also get the asymptotic values for the average Kolmogorov and linear widths of the Sobolev space on the sphere and show that, in the average case setting, the spherical polynomial subspaces are the asymptotically optimal subspaces in the metric, and Fourier partial summation operators and Vallée–Poussin operators are the asymptotically optimal linear operators and are (modulo a constant) as good as optimal nonlinear operators in the metric. [Copyright &y& Elsevier]