International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 1, Pages 11-20
The convolution-induced topology on and linearly dependent translates in
Seminar of Higher Analysis, State University of Ghent, Galglaan 2, GENT B-9000, Belgium
Received 8 October 1980; Revised 7 May 1981
Copyright © 1982 G. Crombez and W. Govaerts. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Given a locally compact Hausdorff group , we consider on the -topology, i.e. the weak topology under all convolution operators induced by functions in . As a major result we characterize the trigonometric polynomials on a compact group as those functions in whose left translates are contained in a finite-dimensional set. From this, we deduce that is different from the -topology on whenever is infinite. As another result, we show that coincides with the norm-topology if and only if is discrete. The properties of are then studied further and we pay attention to the -almost periodic elements of .