Generalized Homogeneity of Means

Generalized Steffensen Means

Homogeneity in generalized function algebras

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 introduc...

generalized steffensen means

Weak homogeneity in generalized function algebras

In this paper, weakly homogeneous generalized functions in the special Colombeau algebras are determined up to equality in the sense of generalized distributions. This yields characterizations that are formally similar to distribution theory. Further, we give several characterizations of equality in the sense of generalized distributions in these algebras.

Correction to "generalized Means"

The purpose of this note is to correct a mistake in the paper entitled Generalized means, which appeared in these Transactions [4]. At the same time, it may be appropriate to report the state of some problems raised there. 1. In §7 of [4] it is asserted that the set of means forms a compact set in the weak operator topology. I am indebted to S. P. Lloyd, of the Bell Telephone Laboratories, for ...