The Haar condition and multiplicity of zeros

Summary In this paper the problem is investigated of how to take the (possibly noninteger) multiplicity of zeros into account in the Haar condition for a linear function space on a given interval. Therefore, a distinction is made between regular and singular points of the interval, and a notion of geometric multiplicity, which always is a positive integer, is introduced. It is pointed out that, for regular zeros (i.e., zeros situate at regular points), aq-fold zero (in the sense that its geometric multiplicity equalsq), counts forq distinct zeros in the Haar condition. For singular zeros (i.e., zeros situated at singular points), this geometric multiplicity has to be diminished by some well-determinable integer.