128 research outputs found

### Subspaces with equal closure

We take a unifying and new approach toward polynomial and trigonometric
approximation in an arbitrary number of variables, resulting in a precise and
general ready-to-use tool that anyone can easily apply in new situations of
interest. The key idea is to show, in considerable generality, that 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. 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 by more general and stronger statements of a
hitherto unknown type. As a side result, we also obtain a new integral
criterion for multidimensional measures to be determinate. At the technical
level, we use quasi-analytic classes in several variables and we show that two
well-known families of one-dimensional weights are essentially equal. The
method can be formulated for Lie groups and this interpretation shows that many
classical approximation theorems are "actually" theorems on the unitary dual of
n-dimensional real space. Polynomials then correspond to the universal
enveloping algebra.Comment: 61 pages, LaTeX 2e, no figures. Second and final version, with minor
changes in presentation. Mathematically identical to the first version.
Accepted by Constructive Approximatio

### Free vector lattices over vector spaces

We show that free vector lattices over vector spaces can be realised in a
natural fashion as vector lattices of real-valued functions. The argument is
inspired by earlier work by Bleier, with some analysis in locally convex
topological vector spaces added. Using this fact for free vector lattices over
vector spaces, we can improve the well-known result that free vector lattices
over non-empty sets can be realised as vector lattices of real-valued
functions. For infinite sets, the underlying spaces for such realisations can
be chosen to be much smaller than the usual ones.Comment: 8 page

### Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights

We prove in a direct fashion that a multidimensional probability measure is
determinate if the higher dimensional analogue of Carleman's condition is
satisfied. In that case, the polynomials, as well as certain proper subspaces
of the trigonometric functions, are dense in the associated L_p spaces for all
finite p. In particular these three statements hold if the reciprocal of a
quasi-analytic weight has finite integral under the measure. We give practical
examples of such weights, based on their classification.
As in the one dimensional case, the results on determinacy of measures
supported on R^n lead to sufficient conditions for determinacy of measures
supported in a positive convex cone, i.e. the higher dimensional analogue of
determinacy in the sense of Stieltjes.Comment: 20 pages, LaTeX 2e, no figures. Second and final version, with minor
corrections and an additional section on Stieltjes determinacy in arbitrary
dimension. Accepted by The Annals of Probabilit

### Paley-Wiener theorems for the Dunkl transform

We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl
transform and prove three instances thereof, one of which involves a limit
transition from Opdam's results for the graded Hecke algebra. Furthermore, the
connection between Dunkl operators and the Cartan motion group is established.
It is shown how the algebra of radial parts of invariant differential operators
can be described explicitly in terms of Dunkl operators, which implies that the
generalized Bessel functions coincide with the spherical functions. In this
context, the restriction of Dunkl's intertwining operator to the invariants can
be interpreted in terms of the Abel transform. Using shift operators we also
show that, for certain values of the multiplicities of the restricted roots,
the Abel transform is essentially inverted by a differential operator.Comment: LaTeX, 26 pages, no figures. References updated and minor changes,
mathematically identical to the first version. To appear in Trans. Amer.
Math. So

### Rapid polynomial approximation in $L_2$-spaces with Freud weights on the real line

The weights $W_\alpha(x)=\exp{(-|x|^{\alpha})}$ $(\alpha>1)$ form a subclass
of Freud weights on the real line. Primarily from a functional analytic angle,
we investigate the subspace of $L_2(\mathbb R, W_\alpha^2(x)\,dx)$ consisting
of those elements that can be rapidly approximated by polynomials. This
subspace has a natural Fr\'echet topology, in which it is isomorphic to the
space of rapidly decreasing sequences. We show that it consists of smooth
functions and obtain concrete results on its topology. For $\alpha=2$ there is
a complete and elementary description of this topological vector space in terms
of the Schwartz functions.Comment: 18 page

### Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system

If $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$,
then a Banach algebra $\ell^1(\Sigma)$ of crossed product type is naturally
associated with this topological dynamical system $\Sigma=(X,\sigma)$. If $X$
consists of one point, then $\ell^1(\Sigma)$ is the group algebra of the
integers.
We study the algebraically irreducible representations of $\ell^1(\Sigma)$ on
complex vector spaces, its primitive ideals and its structure space. The finite
dimensional algebraically irreducible representations are determined up to
algebraic equivalence, and a sufficiently rich family of infinite dimensional
algebraically irreducible representations is constructed to be able to conclude
that $\ell^1(\Sigma)$ is semisimple. All primitive ideals of $\ell^1(\Sigma)$
are selfadjoint, and $\ell^1(\Sigma)$ is Hermitian if there are only periodic
points in $X$. If $X$ is metrisable or all points are periodic, then all
primitive ideals arise as in our construction. A part of the structure space of
$\ell^1(\Sigma)$ is conditionally shown to be homeomorphic to the product of a
space of finite orbits and $\mathbb T$. If $X$ is a finite set, then the
structure space is the topological disjoint union of a number of tori, one for
each orbit in $X$. If all points of $X$ have the same finite period, then it is
the product of the orbit space $X/\mathbb Z$ and $\mathbb T$. For rational
rotations of $\mathbb T$, this implies that the structure space is homeomorphic
to $\mathbb T^2$.Comment: 32 pages. Editorial improvements from the first version, and a few
remarks added. Final version, to appear in Advances in Mathematic

### Positive representations of $C_0(X)$. I

We introduce the notion of a positive spectral measure on a $\sigma$-algebra,
taking values in the positive projections on a Banach lattice. Such a measure
generates a bounded positive representation of the bounded measurable
functions. If $X$ is a locally compact Hausdorff space, and $\pi$ is a positive
representation of $C_0(X)$ on a KB-space, then $\pi$ is the restriction to
$C_0(X)$ of such a representation generated by a unique regular positive
spectral measure on the Borel $\sigma$-algebra of $X$. The relation between a
positive representation of $C_0(X)$ on a Banach lattice and -- if it exists --
a generating positive spectral measure on the Borel $\sigma$-algebra is further
investigated; here and elsewhere phenomena occur that are specific for the
ordered context.Comment: There is now a direct proof of the existence of a generating regular
positive spectral measure in the case of KB-spaces, without resorting to the
Banach space theory. References to the existing literature on the Banach
space case have been added, and perspectives for future research are now
given. 24 pages, to appear in Ann. Funct. Ana

### Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system

If X is a compact Hausdorff space, supplied with a homeomorphism, then a
crossed product involutive Banach algebra is naturally associated with these
data. If X consists of one point, then this algebra is the group algebra of the
integers. In this paper, we study spectral synthesis for the closed ideals of
this associated algebra in two versions, one modeled after C(X), and one
modeled after the group algebra of the integers. We identify the closed ideals
which are equal to (what is the analogue of) the kernel of their hull, and
determine when this holds for all closed ideals, i.e., when spectral synthesis
holds. In both models, this is the case precisely when the homeomorphism has no
periodic points.Comment: 28 page

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