The series development of the quantum-mechanical twisted product is studied. The series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product, either pointwise or in the distributional sense depending on the nature of the factors. A condition is given that ensures convergence and is stronger than previously known results. Possible applications are examined.
Modulo the moment asymptotic expansion, the Cesàro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities, arising from elliptic pseudodifferential operators. We show how Cesàro developments lead to efficient calculations of the expansion coefficients of counting number functionals and Green functions. The bosonic action functional proposed by Chamseddine and Connes can more generally be validated as a Cesàro asymptotic development.
In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If f ∈ S (R) and x 0 ∈ R, and f is locally integrable, then f (x 0 ) = γ distributionally if and only if there exists k such that 1 2π lim x→∞ ax −xf (t)e −ix 0 t dt = γ (C, k) , for each a > 0, and similarly in the case when f is a general distribution. Here (C, k) means in the Cesàro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by lim x→∞ −x≤n≤ax a n e inx 0 = γ (C, k) .We also show that under some extra conditions, as if the sequence {a n } ∞ n=−∞ belongs to the space l p for some p ∈ [1, ∞) and the tails satisfy the estimate ∞ |n|≥N |a n | p = O N 1−p , as N → ∞, the asymmetric partial sums converge to γ . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.