The purpose of this note is to give an extension of the symbolic calculus of Melin for convolution operators on nilpotent Lie groups with dilations. Whereas the calculus of Melin is restricted to stratified nilpotent groups, the extension offered here is valid for general homogeneous groups. Another improvement concerns the$L$^{2}-boundedness theorem, where our assumptions on the symbol are relaxed. The zero-class conditions that we require are of the type $$|D^{\alpha}a(\xi)|\le C_{\alpha}\prod_{j=1}^R\rho_j(\xi)^{-|\alpha_j|},$$ where ρ_{$j$}are “partial homogeneous norms” depending on the variables ξ_{$k$}for$k$>$j$in the natural grading of the Lie algebra (and its dual) determined by dilations. Finally, the class of admissible weights for our calculus is substantially broader. Let us also emphasize the relative simplicity of our argument compared to that of Melin.

In this paper, we give a uniform construction of irreducible genuine characters of the Pin cover W of a Weyl group W, and put them into the context of theory of Springer representations. In the process, we provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of W, and an extended Dirac operator for graded Hecke algebras. We also introduce a q-elliptic pairing for W with respect to the reflection representation V. These constructions are of independent interest. The q-elliptic pairing is a generalization of the elliptic pairing of W introduced by Reeder, and it is also related to S. Kato's notion of (graded) Kostka systems for the semidirect product A W= C [W] S (V).

We define Stasheff polytopes in the spaces of tropical points of cluster A -varieties. We study the supports of products of elements of canonical bases for cluster A -varieties. We prove that, for the cluster A -variety of type A , such supports are Stasheff polytopes.

Varying coefficient models are a useful extension of classical linear models. They arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age. The appeal of these models is that the coef. cient functions can easily be estimated via a simple local regression. This yields a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when different coefficient functions admit different degrees of smoothness. This drawback can be repaired by using our proposed two-step estimation procedure. The asymptotic mean-squared error for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies show that the gain by the two-step procedure can be quite substantial. The methodology is illustrated by an application to an environmental data set.

Differential optical absorption spectroscopy (DOAS) is a powerful tool for detecting and quantifying trace gases in atmospheric chemistry\cite {Platt_Stutz08}. DOAS spectra consist of a linear combination of complex multi-peak multi-scale structures. Most DOAS analysis routines in use today are based on least squares techniques, for example, the approach developed in the 1970s uses polynomial fits to remove a slowly varying background, and known reference spectra to retrieve the identity and concentrations of reference gases. An open problem is to identify unknown gases in the fitting residuals for complex atmospheric mixtures.