# Download PDF by Muhammer Taşkıran, Cünet Kılıç: Analytic Analysis of Lines and Circles (Zambak)

By Muhammer Taşkıran, Cünet Kılıç

ISBN-10: 9752660266

ISBN-13: 9789752660267

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**Example text**

The vector space S ['5",, 5 H] is complete in L ~ ( "d~) [ L 'I ( 'ClG) 1 L q (l)i 4) x L q ("<~~ ~) ] i f and only i f the uniqueness theorem for the corresponding BVP in cA ~ holds. Let us consider the proof in the case of the vector space S. The argument is similar in the remaining two cases. e. =o. G) such that l (4) ßpd6' = 0 r,)~ For lemma VI, (4) implies (3) wi th A• o . Hence u. G ). u. be a function of Jl ~ corresponding to the boundary datum o on '<>G • By lemma VI, (3), with A = 0 , implies (4) and the assumed completeness ß= o Hence, for lemma V, Ll=O.

2. t ~'t iS permitted tobe a closed Jordan curve formed by c•+l arcs meeting at no cusps. In fact it is easily seen that ~(~~J:>L 1 ('dG,). >t:. I 1 +Id1"l . )c1<> i G jJ FICHE RA 38 H~ arcs is supposed a Jordan closed curve formed by C construcexplicit the permits result This cusps. no at meeting tion of a sequence of polynomials (via the "least squares method" in H1 (-aG)] which converges uniformly in G to a given harmonic function, whose boundary values belong to Hi (-;>G). We shall not discuss in this paper ·the method for estimating the approximatio n error relative to the methods which have been surveyed.

Let us consider the proof in the case of the vector space S. The argument is similar in the remaining two cases. e. =o. G) such that l (4) ßpd6' = 0 r,)~ For lemma VI, (4) implies (3) wi th A• o . Hence u. G ). u. be a function of Jl ~ corresponding to the boundary datum o on '<>G • By lemma VI, (3), with A = 0 , implies (4) and the assumed completeness ß= o Hence, for lemma V, Ll=O. The following results were proved in [32]. VIII. For 1. ~ ;1 ~ oo the uniqueness theorem for the Dirichlet problern in Jl-& holds.

### Analytic Analysis of Lines and Circles (Zambak) by Muhammer Taşkıran, Cünet Kılıç

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