Subspaces with Equal Closure |
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Authors: | Email author" target="_blank">Marcel?de?JeuEmail author |
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Institution: | (1) Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands |
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Abstract: | We take a new and unifying approach toward polynomial and trigonometric
approximation in
topological vector spaces used in analysis on R
n
. The idea is to show in
considerable
generality that in such a space a module, which is generated over the polynomials or
trigonometric functions by some set, necessarily has the same closure as the module which is
generated by this same set, but now over the compactly supported smooth functions. The
particular properties of the ambient space or generating set are, to a large degree, irrelevant
for these subspaces to have equal closure. This translation—which goes in fact beyond
modules—allows us, by what is now essentially a straightforward check of a few properties, to
replace many classical results in various spaces by more general statements of a hitherto
unknown type. Even in the case of modules with one generator the resulting theorems on, e.g.,
completeness of polynomials are then significantly stronger than the classical statements. This
extra precision stems from the use of quasi-analytic methods (in several variables) rather than
holomorphic methods, combined with the classification of quasi-analytic weights. In one
dimension this classification, which then involves the logarithmic integral, states that two
well-known families of weights are essentially equal.
As a side result
we also obtain an integral criterion for the determinacy of multidimensional measures which
is less stringent than the classical version.
The approach can be formulated for Lie groups and this interpretation then shows
that many classical approximation theorems are actually theorems on the unitary dual of
R
n
, thus inviting to a change of paradigm. In this interpretation
polynomials correspond to the universal enveloping algebra of R
n
and
trigonometric functions correspond to the group algebra.
It should be emphasized that the point of view, combined with the use of
quasi-analytic methods, yields a rather general and precise ready-to-use tool, which can very
easily be applied in new situations of interest which are not covered by this paper. |
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Keywords: | Approximation Modules over polynomials and trigonometric functions in
several variables Translation invariant subspaces Bernstein problem Lp-Spaces Multidimensional moment problems Multidimensional quasi-analytic classes Quasi-analytic weights Logarithmic integral Harmonic analysis Lie groups |
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