Sentence examples for multiplicity functions from inspiring English sources

We study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group.

We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions m and matrix-valued filter functions H.

Let be a fixed multiplicity function on.

A multiplicity function is a -invariant complex valued function defined on, that is, for all.

As all transpositions are conjugate in, the vector space of multiplicity function is one dimensional.

The Dunkl operator, associated with the Coxeter group and the multiplicity function, is the first-order differential-difference operator.

Let be the Dunkl operator attached to the Coxeter group and the multiplicity function, defined by (see [16]) (1.6).

The Dunkl operators involve a multiplicity function k as parameter [C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989) 167 183].

In the one-dimensional case, the root system is of type, the reflection group, and the multiplicity function is given by a single parameter.

Given a natural number valued function m and a system of functions encoded in a matrix H satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function m and filter system H.

Here is the complex Dunkl Laplacian, and is the complex Dunkl operator attached to the Coxeter group, where is a multiplicity function on and is the reflection with respect to the root.