Banach Journal of Mathematical Analysis


Gabriel Nguetseng and Nils Svanstedt

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We discuss two new concepts of convergence in $L^{p}$-spaces, the so-called weak $\Sigma $-convergence and strong $\Sigma $-convergence, which are intermediate between classical weak convergence and strong convergence. We also introduce the concept of $\Sigma $-convergence for Radon measures. Our basic tool is the classical Gelfand representation theory. Apart from being a natural generalization of well-known two-scale convergence theory, the present study lays the foundation of the mathematical framework that is needed to undertake a systematic study of deterministic homogenization problems beyond the usual periodic setting. A few homogenization problems are worked out by way of illustration.

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Banach J. Math. Anal., Volume 5, Number 1 (2011), 101-135.

First available in Project Euclid: 14 August 2011

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Zentralblatt MATH identifier

Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 35B40: Asymptotic behavior of solutions 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx]

homogenization homogenization algebras \Sigma-convergence Gelfand transformation


Nguetseng, Gabriel; Svanstedt, Nils. \Sigma-convergence. Banach J. Math. Anal. 5 (2011), no. 1, 101--135. doi:10.15352/bjma/1313362985.

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