Homogeneity in generalized function algebras

Abstract

We investigate homogeneity in the special Colombeau algebra on Rd as well as on the pierced space Rd {set minus} {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. © 2007 Elsevier Inc. All rights reserved.