Extensions of Szegö's theory of orthogonal polynomials,III

Let {ø _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofø _n (dμ) when logμ′∈L ¹. In what follows we will discuss the asymptotic behavior of the ratioø _n (dμ ₂)/ø _n (dμ ₁) on the unit circle whendμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=gdμ ₁, where g≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0 for the monic polynomial Φ _n . The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.     

Cyclic sulfonium salts with S ... O interactions 2. Molecular structures with six-membered rings

1-[2-(N-methylcarbamoyl)phenyl]-3H-2,1-naphto-(1,8-d,e)-oxathiin-1-ium chloride (2), 1-[2-(N-methylcarbamoyl)-phenyl]-2-methyl-3-oxo-3H-1, 2-naphto-(1,8-d,e)-thiazin-1-ium chloride (3), 1-[8-(N-methylcarbamoyl)naphtyl]-2-methyl-3-oxo-3H-1, 2-naphto-(1,8-d,e)-thiazin-1-ium chloride (4) and 1-(8-carboxylatonaphtyl)-2-methyl-3-oxo-3H-1,2-naphto-(1,8-d,e)-thiazin-1-ium dipolar ion (5) cyclic sulfonium salts were prepared and their chemical properties investigated (spirosulfurane-formation, hydrolysis). The molecular structures obtained from x-ray diffraction can be described with a considerably distorted trigonal bipyramidal arrangement of the ligands about the sulfonium center, with O/N—S ... O=apical angles of 173.9, 164.9, 156.6, and 159.0°, as well as with S—O/N apical bond lengths of 1.648, 1.671, 1.664, and 1.682 Å. The structures exhibit relatively short S ... O close contacts with interatomic distances of 2.253, 2.448, 2.795, and 2.619 Å.     

On the reaction of dithiocarbamates with nitrogen-containing derivates of oxalic acid

Summary The action of bisimidochlorides of oxalic acid on dithiocarbamates produces 2-thioxo-3-aryl(alkyl)-4,5-diiminothiazolidines by cycloacylation. The molecular structure of 2-thioxo-3-(4-methoxyphenyl)-4,5-bis(phenylimino)-thiazolidine is confirmed by X-ray crystal structure analysis.

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Zur Reaktion von Dithiocarbaminaten mit stickstoffhaltigen Derivaten der Oxalsäure

Zusammenfassung Bei der Einwirkung von Bisimidchloriden der Oxalsäure auf Dithiocarbaminate entstehen durch Cycloacylierung 2-Thioxo-3-aryl(alkyl)-4,5-diiminothiazolidine. Die Molekülstruktur von 2-Thioxo-3-(4-methoxyphenyl)-4,5-bis(phenylimino)thiazolidin wird durch Röntgenkristallstrukturanalyse bestätigt.

     

Subexpansions,superexpansions and uniqueness properties in non-integer bases

The β-expansions, i.e., greedy expansions with respect to non-integer bases q>1, were introduced by Réenyi and then investigated by many authors. Some years ago, Erdős, Horváth and Joó found the surprising fact that there exist infinitely many numbers 1<q<2 for which the β-expansion of 1 is the unique possible expansion with coefficients 0 or 1. Subsequently, the unique expansions were characterized in [9] and this characterization led to the determination (in [17]) of the smallest number q having this curious property. It is intimately related to the classical Thue-Morse sequence. Allouche and Cosnard recently proved that this q is transcendental. The purpose of this paper is to extend the previous results for expansions in arbitrary non-integer bases q>1. We also determine the smallest q having the corresponding uniqueness property in each case, and we prove that all of them are transcendental. We will also obtain some probably new properties of the Thue-Morse sequence. In the last section we answer a question concerning the existence of universal expansions, a notion introduced in [12]. This revised version was published online in June 2006 with corrections to the Cover Date.     

Weighted polynomial inequalities

For the weights exp (?|x|^λ), 0<λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0<λ<1. The proof is based on the solution of the problem of how fast a polynomialP _n can decrease on [?1,1] ifP _n (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedL ^p spaces.