A spectral Paley-Wiener theorem

The Fourier inversion formula in polar form is \(f(x) = \int_0^\infty {P_\lambda } f(x)d\lambda \) for suitable functionsf on ? ⁿ , whereP _λ f(x) is given by convolution off with a multiple of the usual spherical function associated with the Euclidean motion group. In this form, Fourier inversion is essentially a statement of the spectral theorem for the Laplacian and the key question is: how are the properties off andP _λ f related? This paper provides a Paley-Wiener theorem within this avenue of thought generalizing a result due to Strichartz and provides a spectral reformulation of a Paley-Wiener theorem for the Fourier transform due to Helgason. As an application we prove support theorems for certain functions of the Laplacian.     

Determination of the sulfur components of gasoline streams by capillary column gas chromatography with sulfur chemiluminescence detection

A method using a sulfur chemiluminescence detector (SCD) has been developed for the qualitative and quantitative determination of sulfur components in stabilized gasoline-range process streams, including blended gasolines containing between 1 and > 4000 ppm total sulfur. The detection limit per sulfur component was approximately 50 ppb. On-column injection was employed for optimum precision and accuracy. A new probe material was found which did not soften under FID operating conditions. A new method, using a hydrogen sulfide permeation tube, was developed for rapid alignment of the probe in the FID.     

(Z)-3-p-tolylsulfinylacrylonitrile as a chiral dienophile: diels-alder reactions with furan and acyclic dienes

The behavior of (Z)-3-p-tolylsulfinylacrylonitrile (1) as a chiral dienophile has been evaluated from its reactions with furan and acyclic dienes. Electrostatic interactions of the cyano group with the sulfinyl one restrict the conformational mobility around the C-S bond, thus controlling the pi-facial selectivity, which is almost complete in all cases, the approach of the diene from the less-hindered face of the dienophile (that bearing the lone electron pair) in the predominant rotamer being the favored one. The regioselectivity is also completely controlled by the cyano group. Additionally, the reactivity of compound 1 as well as its endo-selectivity are both higher than those observed for the corresponding (Z)-3-sulfinylacrylates, thus proving the potential of sulfinylnitriles as chiral dienophiles.