Mario Ullrich, Tino Ullrich, Van Kien Nguyen,
"Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube"
, in Constructive Approximation, Vol. 46, Nummer 1, Seite(n) 69?108, 2017, ISSN: 1432-0940
Original Titel:
Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube
Sprache des Titels:
Englisch
Original Kurzfassung:
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.
Forschungs-