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Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.46698/g872857834755h Quasianalyticity Criterion of SalinasKorenblyum Type for Convex Domains
Gaysin, R. A.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 3.
Abstract:
The quasianalyticity problem of the class \(C_{I}(M_n)\) for interval \(I\) is known to be solved by the DenjoyCarleman theorem. It follows from wellknown Men'shov example that not only this theorem but the very statement of the quasianalyticity problem of the class \(C_{K}(M_n)\) doesn't expand on the case of arbitrary continuum \(K\) of the complex plain. The quasianalyticity problem was studied for Jordan domains and rectifiable arcs including quasismooth arcs by a number of authors. We discuss in this article theorems of DenjoyCarleman type in the convex domains of the complex plane, more precisely, connection between R. S. Yulmukhametov criterion of quasianalyticity of the Carleman class \(H(D,M_n)\) for arbitrary convex domain \(D\) and R. Salinas criterion for the class \(H(\Delta_{\alpha},M_n)\) with angle \(\Delta_{\alpha}=\{z: \arg z\leq\frac{\pi}{2}\alpha,\ \ 0<\alpha\leq1\}\). The problem of quasianalyticity of the class \(H(D,M_n)\) is to find necessary and sufficient conditions for sequence \(M_n\) and point \(z_0\in\partial D\) for quasianalyticity of the class \(H(D,M_n)\) at this point. The answer to question of simultaneous quasianalyticity or nonquasianalyticity these Carleman classes at a point \(z=0\) has been obtained in therms of special integral condition which characterizes the degree of proximity of the domain boundaries \(D\) and the angle \(\Delta_{\alpha}\) in the neighbourhood of origin. Geometric interpretation of this integral condition and explicit examples illustrating essentiality of this condition are given.
Keywords: Carleman class, convex domain, Salinas criterion, integral condition of local aboutness of the boundaries
Language: Russian
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For citation: Gaysin, R. A. Quasianalyticity Criterion of SalinasKorenblyum Type for Convex Domains,
Vladikavkaz Math. J., 2020, vol. 22, no. 3, pp. 5871.
DOI 10.46698/g872857834755h ← Contents of issue 
 

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