Polynomial approximation on the boundary and strictly inside

We investigate the possibility of approximating a function on a compact setK of the complex plane in such a way that the rate of approximation is almost optimal onK, and the rate inside the interior ofK is faster than on the whole ofK. We show that ifK has an external angle smaller than π at some point z_o∈δK, then geometric convergence insideK is possible only for functions that are analytic at z_o. We also consider the possibility of approximation rates of the form exp(?cn ^β), for approximation insideK, where β is related to the largest external angle ofK. It is also shown that no matter how slowly the sequence {γ _n } tends to zero, there is aK and a Lip β, β<1, functionf such that approximation insideK cannot have order {γ _n }.     

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.     

Sharp Nikolskii inequalities with exponential weights

w _a(x)=exp(–x^a), xR, a0. , N _n (a,p,q) — (2), _n P _nw_ap, CN_n(a,p, q)P_nw_aq. , — , {P _n}, .

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This material is based upon research supported by the National Science Foundation under Grant No. DMS-84-19525, by the United States Information Agency under Senior Research Fulbright Grant No. 85-41612, and by the Hungarian Ministry of Education (first author). The work was started while the second author visited The Ohio State University between 1983 and 1985, and it was completed during the first author's visit to Hungary in 1985.     

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.