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Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube. (English) Zbl 1376.65028
This paper deals with the optimal approximation of multivariate integrals \(\int_{[0,1]^d} f(x)\,{\mathrm d}x\) of nonperiodic/periodic integrands \(f\) via Frolov cubature formulae. The authors show that the asymptotic order of the minimal worst-case integration error is not affected by boundary conditions in Besov spaces and Triebel-Lizorkin spaces of functions with dominating mixed smoothness. As essential tool of this research, the Besov spaces and Triebel-Lizorkin spaces of dominating mixed smoothness are characterized by mixed iterated differences. The authors present two modified Frolov cubature formulae that yield the same order of convergence up to a constant. This constant involves the norms of a change of variable mapping and a pointwise multiplication mapping between function spaces of dominating mixed smoothness. In the main part of this paper, the authors present new results on boundedness of change of variable mapping and pointwise multiplication mapping between function spaces of dominating mixed smoothness.

MSC:
65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B38 Linear operators on function spaces (general)
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