Van Kien Nguyen

Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube

In a recent article, two of the present authors studied Frolov’s cubature formulae and their optimality in Besov–Triebel–Lizorkin spaces of functions with dominating mixed smoothness supported in the unit cube. In this paper, we give a general result that the asymptotic order of the minimal worst-case integration error is not affected by boundary conditions in the above mentioned spaces. In fact, we propose two tailored modifications of Frolov’s cubature formulae suitable for functions supported on the cube (not in the cube) that yield the same order of convergence up to a constant. This constant involves the norms of a “change of variable” and a “pointwise multiplication” mapping, respectively, between the function spaces of interest. We complement, extend, and improve classical results on the boundedness of change of variable mappings in Besov–Sobolev spaces of mixed smoothness. The second modification, suitable for classes of periodic functions, is based on a pointwise multiplication and is therefore most likely more suitable for applications than the (traditional) “change of variable” approach. These new theoretical insights are expected to be useful for the design of new (and robust) cubature rules for multivariate functions on the cube.

Weyl numbers of embeddings of tensor product Besov spaces

In this paper we investigate the asymptotic behaviour of Weyl numbers of embeddings of tensor product Besov spaces into Lebesgue spaces. These results will be compared with the known behaviour of entropy numbers.

Gelfand numbers of embeddings of mixed Besov spaces

Gelfand numbers represent a measure for the information complexity which is given by the number of information needed to approximate functions in a subset of a normed space with an error less than

Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness

\varepsilon

Pointwise multipliers for Besov spaces of dominating mixed smoothness - II

. More precisely, Gelfand numbers coincide up to the factor 2 with the minimal error

Isotropic and dominating mixed Lizorkin–Triebel spaces — a comparison

e^{\rm wor}(n,\Lambda^{\rm all})

Bernstein numbers of embeddings of isotropic and dominating mixed Besov spaces

which describes the error of the optimal (non-linear) algorithm that is based on

On a problem of Jaak Peetre concerning pointwise multipliers of Besov spaces

n

Pointwise multipliers for Sobolev and Besov spaces of dominating mixed smoothness

arbitrary linear functionals. This explain the crucial role of Gelfand numbers in the study of approximation problems. Let

Monte Carlo methods for uniform approximation on periodic Sobolev spaces with mixed smoothness

S^t_{p_1,p_1}B((0,1)^d)