Homogeneity in generalized function algebras |
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Authors: | Clemens Hanel Stevan Pilipovi? |
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Institution: | a Faculty of Mathematics, University of Vienna, Nordbergstraße 15, 1090 Vienna, Austria b Faculty of Sciences and Mathematics, University of Novi Sad, Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia c Faculty of Civil Engineering, University of Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria |
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Abstract: | We investigate homogeneity in the special Colombeau algebra on Rd as well as on the pierced space Rd?{0}. It is shown that strongly scaling invariant functions on Rd are simply the constants. On the pierced space, strongly homogeneous functions of degree α admit tempered representatives, whereas on the whole space, such functions are polynomials with generalized coefficients. We also introduce weak notions of homogeneity and show that these are consistent with the classical notion on the distributional level. Moreover, we investigate the relation between generalized solutions of the Euler differential equation and homogeneity. |
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Keywords: | Generalized functions Homogeneity Scaling invariance Colombeau algebras |
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