By Daniel Carando, Andreas Defant, Pablo Sevilla-Peris
http://www.sciencedirect.com/science/article/pii/S0022123615003870
We introduce non-linear types of the classical cotype of Banach areas. We exhibit that areas with l.u.st. and cotype, and areas having Fourier cotype take pleasure in our non-linear cotype. We practice those recommendations to get effects on convergence of vector-valued energy sequence in countless many variables and on ℓ1-multipliers of vector-valued Dirichlet sequence. ultimately we introduce cotype with appreciate to indexing units, an concept that incorporates our prior definitions.
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20) where D is a domain, S is its boundary, and N is the unit outer normal to 5. Finding u from the data {/, h} is an ill-posed problem: Small perturbation of the data {/, h) may lead to the pair {fs,hs}, which does not correspond to any harmonic function in D. 20) cannot be chosen arbitrarily. 20) uniquely. Therefore / determines h uniquely as well. The map A:f^h is called the Dirichlet-to-Neumann map. This map is injective and its properties are known. 7 Minimization problems. Let f(u) >m> - o o be a continuous functional in a Banach space X.
One of the basic motivations for the theory of local tomography, originated in [116] and [117], and developed in [118], was the desire to minimize the X-ray radiation of patients. , see [118, p. 31]): Here f(a,p) = f(-a,-p), VpeR, Sl is the unit sphere in R 2 , that is, the set \a\ = 1, a € R 2 , and fp = -A. ) with compact support. 44) means that one has to know f(a,p) for all a € S1 and all p 6 R in order to recover f(x) at the point x. A. G. Ramm has posed the following question: / / one cannot recover / ( x ) from local tomographic data, what practically useful information can one recover from these data?
69) exists and is unique. 69) is quadratic, so a necessary condition for its minimizer is a linear equation, the Euler equation. Assume first that A is bounded. 71) A*Au + au = A*fs. 72) Ta:T + ai. 71) has a solution u a ä = T~1A*f$ and this solution is unique. {ν-α,δ + v), Vu € H, and the equation sign is attained if and only if v = 0. Let us choose a = α(δ) so that us := ua(<5),5 would satisfy the relation lim || U i - y|| = 0. 74) a—»U To do so, we estimate: | | Γ - Μ · / « - y|| < \\T~lA*{fs - f)\\ + \\T~lA*f - y\\ := Λ + J 2 .