Product formulas and convolution structure for Fourier-Bessel series

One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter ,n=0, 1, 2,, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases and . As a consequence, a positive convolution structure is established for . The method of proof is based on solving a hyperbolic initial boundary value problem.Communicated by Tom H. Koornwinder.     

Heat “polynomials” for a singular differential operator on (0, ∞)

We generalize the theory of the heat polynomials introduced by P. V. Rosenbloom and D. V. Widder for a more general class of singular differential operator on (0, ). The heat polynomials associated with the Bessel operator and studied by D. T. Haimo appear as a particular case in this paper. In the special cases of second derivative and Bessel operators the heat polynomials are in fact polynomials inx andt, however, this property does not hold in general.Communicated by Tom. H. Koornwinder.     

Solutions rationnelles de certaines équations fonctionnelles

Dans cet article, nous démontrons essentiellement les deux résultats suivants, qui montrent que les solutions séries formelles à coefficients dans de certaines équations fonctionnelles sont rationnelles. Soient tout d'abords un entier naturel non nul, eta _i ,b _i ,(i = 1, , s), 2s nombres complexes, lesa _i étant non nuls. On définit l'ensembleA comme étant l'intersection des parties de , contenant l'origine et stables par toutes les applicationsg _i (x) = a _i x + b _i . On a alors le résultat suivant: Théorème 1.Soient f, R ₁, ,R _s s + 1 fractions rationnelles de (x), régulières à l'origine, et a_i, b_i (i = 1,, s), 2s éléments de . On suppose que les a_i sont non nuls et de module strictement inférieur à un pour tout i = 1,, s. Soit y(x) un élément de [[x]], vérifiant l'équation fonctionnelle

Interpolatory properties of best rationalL 1-approximations

In this paper the distribution of the zeros of the error function for bestL ₁-approximation by rational functions fromR _n,m is considered. It is shown that the maximal distance between such zeros isO(1/(n–m)), ifn > m.Communicated by Edward B. Saff.     

A proof of Freud's conjecture for exponential weights

LetW (x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ²(x) are finite. Let {p _n (W ²;x)} ₀ denote the sequence of orthonormal polynomials with respect to the weightW ²(x), and let {A _n } ₁ and {B _n } ₁ denote the coefficients in the recurrence relation

(甲基)丙烯酸酯的基团转移聚合

<正> 基团转移聚合方法自问世以来,由于其突出的优点,受到日益广泛的重视。本文来用丙二酸二乙酯的硅烯醇醚类引发剂3-乙氧基-3-三甲基硅氧基丙烯酸乙酯 (CH_3CH_2O_2CCH=C(OCH_2CH_3)OSiMe_3)和亲核型催化剂四乙基二氯化氢氨((CH_3CH_2)_4NHF_2)进行甲基丙烯酸甲酯(MMA)、丙烯酸丁酯(BA)、丙烯酸乙酯(EA)和丙烯酸甲酯(MA)的基因转移聚合。得到了近     

Spectral studies of bridged bisindenylmetallocenes and products of their reactions with polymethylalumoxane in toluene and dichloromethane in the visible region

Quantitative characteristics of changes in the energy of charge transfer from -bonded ligands to metal are presented for the series of structurally similar ansa-metallocene complexes of IVB Group elements. The changes are caused by the solvation effect, replacement of -bonded chlorine ligands by methyl groups and of bridging ethylene groups by dimethylsilylene moieties, introduction of the methyl substituent into position 2 and phenyl substituent into position 4 of the indenyl system, variation of the transition metal in metallocene, and formation of complexes with polymethylalumoxane (MAO) at different Al_MAO/Zr ratios. These effects are found to be additive.     

In search of 1,3-disila/germa/stannabicyclo[1.1.1]pentanes with short bridgehead-bridgehead distances and low ring strain energies

The strain energies and through-space distances between the two bridgehead E atoms of a selection of 1,3-dimethyl-1,3-ditetrelbicyclo[1.1.1]pentanes (tetrel E = Si, Ge or Sn) were examined by quantum chemical calculations at MP2 and B3LYP levels. The aim is to identify which bridges lead to short through-space E,E distances, and simultaneously, to as low strain as possible. A short E,E distance should improve through-space interaction, and a low strain should promote the thermal stability and possibly also facilitate their synthesis. The bridges examined included CH₂, CMe₂, CtBu₂, C(CH₂)n (n = 2–4), O, NMe, S, PMe, SiMe₂, GeMe₂, and SnMe₂. The calculations indicate that the phospha bridge is a good compromise providing reasonably low strain as well as E,E through-space distances which are only longer than normal E–E single bonds by factors of 1.06–1.10. This paper is dedicated to Professor Mitsuo Kira in recognition of his stimulating Si chemistry and his 2005 Wacker Award.     

基团的电子效应与单取代苯对位^1H,^13C的化学位移

基团的电子效应与单取代苯对位１Ｈ、１３Ｃ的化学位移＊韩长日冯娇杨钟照平（海南师范学院化学系海口５７１１５８）关键词１Ｈ的化学位移１３Ｃ的化学位移基团电负性共轭效应引言在１Ｈ、１３Ｃ核磁共振谱中，化学位移值的大小主要取决于屏蔽作用的大小，而屏蔽作用的大...     

Synthesis and crystal structures of Group 6 metal carbonyl complexes containing S-rich bis(pyrazol-1-yl)methane ligands

The reaction of 3(5)-methylthio-5(3)-phenylpyrazole with dibromomethane under phase-transfer catalytic conditions only affords a new ligand, bis(3-phenyl-5-methylthiopyrazol-1-yl)methane. However, the reaction of 3(5)-methylthio-5(3)-p-methoxyphenylpyrazole or 3(5)-methylthio-5(3)-tert-butylpyrazole with dibromomethane under the same conditions yields three isomers, respectively, indicating that the substituents significantly affect the steric and electronic properties of pyrazole ring during the formation of ligands. Treatment of these potential polydentate ligands with M(CO)₆ (M=Cr, Mo or W) under UV irradiation at room temperature affords (NN)M(CO)₄ derivatives, in which some complexes contain asymmetric substituted bis(pyrazol-1-yl)methane ligands. The X-ray crystal structure analyses indicate that the sulfur atoms in these complexes do not take part in the coordination to the metal centers, and S-rich bis(pyrazol-1-yl)methanes actually act as bidentate chelating ligands by two nitrogen atoms. It is also interesting that in order to reduce the repulsion of methyl groups with carbonyls, the methyl groups in these complexes are oriented away from the metal centers.