Interpolating refinable function vectors with reflection properties

In this paper, we investigate interpolating refinable function vectors whose component functions constitute reflection pairs. A necessary condition for an interpolating refinable function vector to satisfy the reflection property is provided for any dilation factor and multiplicity. Although, the converse is not true in general, for some special cases it becomes a sufficient condition as well. We concentrate on the relation between a dilation factor and multiplicity, and provide the necessary and sufficient condition on the coefficients of its refinement mask in order for such refinable function vector to form a reflection pair. Finally, trivial solutions consisting of translated versions of one component function which is symmetric by itself are discussed. To illustrate the results, various numerical examples are provided.