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

Peak recognition technique by a computer program copying the human judgement

Summary It is still difficult to determine peaks and peak boundaries properly, though peak recognition is very important for the precision of quantitative data. A new computer program overcomes these problems using a method which is adapted from human judgements. The algorithm was developed for HPLC but can also be used in other fields of analytical chemistry.     

一种利用质谱信息进行结构特征鉴定的模式识别方法

本文提出了分段式离子系列谱编码方法,实验证明具有很好的结构表征能力。本程序主要是以模式识别技术为基础,实现了对未知物谱图的KNN分类,进而得到未知物的主要结构特征。这个程序应用于有机化合物的分类,取得了满意的结果。     

Multivariate classification based on metal contents in human senile cataract lenses

Multivariate classification methods were used to evaluate data on the concentrations of eight metals in human senile lenses measured by atomic absorption spectrometry. Principal components analysis and hierarchical clustering separated senile cataract lenses, nuclei from cataract lenses, and normal lenses into three classes on the basis of the eight elements. Stepwise discriminant analysis was applied to give discriminant functions with five selected variables. Results provided by the linear learning machine method were also satisfactory; the k-nearest neighbour method was less useful.     

Repeatability and pattern recognition of bacterial fatty acid profiles generated by direct mass spectrometric analysis of in situ thermal hydrolysis/methylation of whole cells

Direct CI mass spectrometry profiling of fatty acid methyl esters (FAMEs) from in situ thermal hydrolysis/methylation (THM) of whole bacterial cells with tetramethylammonium hydroxide (TMAH) has been demonstrated as a potential method for real time and fieldable detection/identification of microorganisms. Bacillus anthracis (Ames), Yersinia pestis (Nair. Kenya), Vibrio cholerae (E1 Tor), Brucella melitensis (Abortus wild) and Francisella tularensis (LVS vaccine) were profiled by this method during a 10-month period. Repeatability of the in situ FAME data was calculated using one-way analysis of variance (ANOVA) and a t-test. Artificial neural network (ANN) and multivariate statistics of the FAME profiles were also compared for bacterial identification/classification. Equivalent results were obtained with a multivariate rule building expert system (MuRES) and the ANN. However, the ANN analysis required much less computer time and was deemed the best choice for this application. In situ THM FAME profiles of the bacterial samples provided comparable results with those obtained from the Microbial Identification System (MIDI) (Newark, DE) wet chemistry-gas chromatographic based system.