32 research outputs found
Divided Differences & Restriction Operator on Paley-Wiener Spaces for Carleson Sequences
For a sequence of complex numbers we consider the restriction
operator defined on Paley-Wiener spaces
(). Lyubarskii and Seip gave necessary and sufficient conditions on
for to be an isomorphism between and a
certain weighted space. The Carleson condition appears to be necessary.
We extend their result to Carleson sequences (finite unions of disjoint
Carleson sequences). More precisely, we give necessary and sufficient
conditions for to be an isomorphism between and
an appropriate sequence space involving divided differences
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Intramolecular O -> Te and N -> Te coordination bonds in molecules of beta-tellurocyclohexenals and their nitrogen analogs
The structures of beta-tellurocyclohexenals and their nitrogen analogs, viz., beta-methyl-tellurocyclohexenal (6), beta-(4-ethoxyphenyltelluro)cyclohexenal (7), di(2-formylcyclohexen-1-yl) telluride (8), beta-(4-ethoxyphenyltelluro)cyclohexenylidene(4'-methylaniline) (9), beta-bromotellurenylcyclohexenylidene(4'-methylaniline) (10), and beta-bromotellurenylcyclohexenal (4-methylbenzoyl)hydrazone (11), were studied by X-ray diffraction analysis. Compounds 6-11 have a Z configuration at the double bond, which provides the formation of intramolecular O-->Te or N-->Te coordination bonds. The bonds about the Te atom have a T-shaped configuration. There is only one of two possible O-->Te coordination bonds in dialdehyde 8 and, consequently, this compound belongs to the 10-Te-3-tellurane structural type. Hydrazone 11 possesses both N-->Te and O-->Te intramolecular coordination bonds. Taking into account these interactions, the coordination polyhedron of the tellurium atom can be considered as a trigonal bipyramid. The intramolecular O-->Te or N-->Te coordination bond lengths in compounds 6 (2.692 Angstrom), 7 (2.657 Angstrom), 8 (2.657 Angstrom), and 9 (2.690 Angstrom) are 0.9-1.0 Angstrom smaller than the sums of the van der Waals radii of the corresponding atoms. These bond lengths in compounds 10 (2.170 Angstrom) and 11 (2.203 Angstrom) are almost equal to the standard covalent bond length
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Synthesis and structure of N-arylimines of beta-tellurocyclohexenals with the intramolecular coordination N -> Te bonds
A series of N-arylimines of beta-tellurocyclohexenals 11 have been synthesized and the molecular and crystal structures of the compounds 11a-e and also beta-(dimethyltelluronium)cyclohexenal perchlorate 12 studied by X-ray crystallography. All the compounds contain strong intramolecular coordination N --> Te (O --> Te) bonds of the hypervalent type. In 11a-e, the lengths of the N --> Te bonds are within the range of 2.690-2.147 Angstrom and are 1.0-1.5 Angstrom shorter than the sum of the van der Waals radii of respective atoms. In the N-arylimines 11a-e with the electronegative groups attached to the tellurium center, the lengths of the N --> Te bonds are very close to that characteristic of a standard covalent N-Te bond. The experimental observed geometries are well reproduced by the DFT calculations performed at B3LYP/LanL2DZ level of approximation. The energies of the intramolecular coordination N --> Te bonds vary from 23 kJ mol(-1) for 11a to 119 kJ mol(-1) for 11e. The calculated energy of the O --> Te bond in 12 was found to be 50 kJ mol(-1). The Te-125 NMR chemical shifts of compounds 11 span the wide range of 734.3-1622.4 ppm. The largest downfield Te-125 NMR chemical shifts are observed in the case of the compounds 11e, f in which the most electronegative atoms are attached to the tellurim centers