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Zbl 1229.46035
Nguetseng, Gabriel; Svanstedt, Nils
$\Sigma $-convergence. (English)
[J] Banach J. Math. Anal. 5, No. 1, 101-135, electronic only (2011). ISSN 1735-8787/e

The usual approach to the weak convergence of a product of functions is to require one of the sequences to converge strongly. For example, if $u_n \to u$ in $L^p(\Omega)$, $v_n \rightharpoonup v$ in $L^{p^{\prime}}(\Omega)$, then $u_n v_n \rightharpoonup uv$ in $L^1(\Omega)$, where $\rightharpoonup$ means weak convergence. The authors generalize this by considering weak $\Sigma$-convergence, $\Sigma$-convergence, homogenization algebras and the Gelfand transform. The details are rather technical (see the paper). It is a generalization of the two scale convergence of [{\it G. Nguetseng}, SIAM J. Math. Anal. 20, No. 3, 608--623 (1989; Zbl 0688.35007)] to the non-periodic setting, using et al the Gelfand transform. The authors illustrate the techniques by applying the ideas to a few homogenization problems.
[Raymond Johnson (Houston)]

MSC 2000:: *46J10 Banach algebras of continuous functions
35B27 Homogenization, etc.
28A20 Measurable and nonmeasurable functions

Keywords: homogenization; homogenization algebras; $\Sigma $-convergence; Gelfand transformation

Citations: Zbl 0688.35007

Cited in: Zbl pre06099665

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