We consider the Gelfand and Kolmogorov numbers of compact embeddings between weighted function spaces of Besov and Triebel-Lizorkin type with polynomial weights in the non-limiting case. Our main purpose here is to complement our previous results in [32] in the context of the quasi-Banach setting, 0 < p, q ≤ ∞. In addition, sharp estimates for their approximation numbers in several cases left open in Skrzypczak (2005) [25] are provided.
In this paper, we study the complexity of information of approximation problem on the multivariate Sobolev space with bounded mixed derivative MW r p,α (T d ), 1 < p < ∞, in the norm of Lq(T d ), 1 < q < ∞, by adaptive Monte Carlo methods. Applying the discretization technique and some properties of pseudo-s-scale, we determine the exact asymptotic orders of this problem. Keywords: adaptive Monte Carlo method, Sobolev space with bounded mixed derivative, asymptotic order MSC(2000): 41A46, 41A63, 65C05, 65D99
We study the information-based complexity of the approximation problem on the multivariate Sobolev space with bounded mixed derivative MW r p, in the norm of L q by linear Monte Carlo methods. Applying the Maiorov's discretization technique and some properties of pseudo-s-scale, we determine the exact orders of this problem for 1 < p, q < ∞.
Denote by B 2σ,p (1 < p < ∞) the bandlimited class p-integrable functions whose Fourier transform is supported in the interval [−σ, σ]. It is shown that a function in B 2σ,p can be reconstructed in L p (R) by its sampling sequences {f (kπ/σ)} k∈Z and {f (kπ/σ)} k∈Z using the Hermite cardinal interpolation. Moreover, it will be shown that if f belongs to L r p (R), 1 < p < ∞, then the exact order of its aliasing error can be determined.
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