Tractability of Approximation for Some Weighted Spaces of Hybrid Smoothness

A great deal of work has studied the tractability of approximating (in the L 2-norm) functions belonging to weighted unanchored Sobolev spaces of dominating mixed smoothness of order 1 over the unit d-cube. In this paper, we generalize these results. Let r and s be non-negative integers, with r ≤ s. We consider the approximation of complex-valued functions over the torus 𝕋 d = [ 0 , 2 π ] d $$\mathbb {T}^d=[0,2\pi ]^d$$ from weighted spaces H Γ s , 1 ( 𝕋 d ) $$H^{s,1}_\varGamma (\mathbb {T}^d)$$ of hybrid smoothness, measuring error in the H r ( 𝕋 d ) $$H^r(\mathbb {T}^d)$$ -norm. Here we have isotropic smoothness of order s, the derivatives of order s having dominating mixed smoothness of order 1. If r = s = 0, then H 0 , 1 ( 𝕋 d ) $$H^{0,1}(\mathbb {T}^d)$$ is a well-known weighted unachored Sobolev space of dominating smoothness of order 1, whereas we have a generalization for other values of r and s. Besides its independent interest, this problem arises (with r = 1) in Galerkin methods for solving second-order elliptic problems. Suppose that continuous linear information is admissible. We show that this new approximation problem is topologically equivalent to the problem of approximating H Γ s − r , 1 ( 𝕋 d ) $$H^{s-r,1}_\varGamma (\mathbb {T}^d)$$ in the L 2 ( 𝕋 d ) $$L_2(\mathbb {T}^d)$$ -norm, the equivalence being independent of d. It then follows that our new problem attains a given level of tractability if and only if approximating H Γ s − r , 1 ( 𝕋 d ) $$H^{s-r,1}_\varGamma (\mathbb {T}^d)$$ in the L 2 ( 𝕋 d ) $$L_2(\mathbb {T}^d)$$ -norm has the same level of tractability. We further compare the tractability of our problem to that of L 2 ( 𝕋 d ) $$L_2(\mathbb {T}^d)$$ -approximation for H Γ 0 , 1 ( 𝕋 d ) $$H^{0,1}_\varGamma (\mathbb {T}^d)$$ . We then analyze the tractability of our problem for various families of weights.