  In a recent article by two of the present authors it turned out that Frolov's
cubature formulae are optimal and universal for various settings
(Besov-Triebel-Lizorkin spaces) of functions with dominating mixed smoothness.
Those cubature formulae go well together with functions supported inside the
unit cube $[0,1]^d$. The question for the optimal numerical integration of
multivariate functions with non-trivial boundary data, in particular
non-periodic functions, arises. In this paper we give a general result that the
asymptotic rate 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) which provide the same minimal
worst-case error 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. In fact, we complement,
extend and improve classical results by Bykovskii, Dubinin and Temlyakov on the
boundedness of change of variable mappings in Besov-Sobolev spaces of mixed
smoothness. Our proof technique relies on a new characterization via integral
means of mixed differences and maximal function techniques, general enough to
treat Besov and Triebel-Lizorkin spaces at once. The second modification, which
only tackles the case 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.
