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 L2L_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 C0(X)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