flamboyant
发表于 2025-3-23 10:52:56
Generalization of One-Dimensional Carleman FormulasLet . be a bounded domain in ℂ. with piecewise smooth boundary .. Consider a function ƒ ε .(.) and a set . ⊂ . of positive Lebesgue measure and assume that the .( .(.))-convex hull of. does not contain the coordinate origin.Then there is a function . ∊ .(.(.)) such that .(0) = 1 and
珍奇
发表于 2025-3-23 15:07:42
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Chronological
发表于 2025-3-23 18:40:28
Carleman Formulas in Homogeneous DomainsLet . be a classical domain in ℂ., and . its distinguished boundary (Shilov boundary). We define the Hardy classes .(.) as follows: a function . ∈ .(.) belongs to the class HP(D), 0 < . < ∞, if
下垂
发表于 2025-3-23 23:23:50
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phytochemicals
发表于 2025-3-24 04:41:37
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要素
发表于 2025-3-24 09:30:35
https://doi.org/10.1007/978-3-642-41461-9hy formula (see ).is valid. Let us consider on the boundary . a measurable set . of positive Lebesgue measure. The problem is to reconstruct .(.) in . from its values not on the whole boundary . as in (1.1) but on . ⊂ . only. Applying a simple, but very fruitful idea of Carleman we cons
支形吊灯
发表于 2025-3-24 13:02:25
https://doi.org/10.1007/978-3-642-41461-9s involving integration over the whole boundary . of a domain . an integral representation involving integration over a set . ⊂ ., rests on the availability of a function .(.) of class .(.) satisfying two conditions (see sec. 1):
套索
发表于 2025-3-24 18:10:32
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jocular
发表于 2025-3-24 22:35:56
https://doi.org/10.1007/978-3-642-41461-90 instead of —. (see the remark in sec. 30). All computations were made with double precision. The first to be considered was the simple function . = (. -2 -2.). ,belonging to the Hardy class . ,and points . from the interval . For functions of this class formula (30.4) (with . instead of -.)
戏服
发表于 2025-3-24 23:09:53
https://doi.org/10.1007/978-3-642-41461-9ni formula for the case when M is an arc with ends on real axis. But if M = Γ is an arc in the unit disk with ends on the unit circle then we can give a simpler formula (see the beginning of example 3, sec. 1):