On Ascertaining Inductively the Dimension of the Joint Kernel of Certain Commuting Linear Operators

✍ Scribed by Carl de Boor; Amos Ron; Zuowei Shen

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 348 KB
• Volume: 17
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.1996.0011

✦ Synopsis

Given an index set X, a collection ‫ނ‬ of subsets of X all of the same cardinality , Ä 4 and a collection l of commuting linear maps on some linear space, the

x xg X Ž . family of linear operators whose joint kernel K s K ‫ނ‬ is sought consists of all l [ Ł l with A any subset of X which intersects every B g ‫.ނ‬ The A a g A a Ž . goal is to establish conditions, on ‫ނ‬ and l, which ensure that dim K ‫ނ‬ s ŽÄ 4. Ý dim K B , or, at least, one or the other of the two inequalities contained B g ‫ނ‬ in this equality. Concrete instances of this problem arise in box spline theory, and specific conditions on l were given by Dahmen and Micchelli for the case that ‫ނ‬ consists of the bases of a matroid. We give a new approach to this problem and establish the inequalities and the equality under various rather weak conditions on ‫ނ‬ and l. These conditions involve the solvability of certain linear systems of the form l ?s , b g B, with B g ‫,ނ‬ and the existence of ''placeable'' elements of b b X , i.e., of x g X for which every B g ‫ނ‬ not containing x has all but one element in common with some BЈ g ‫ނ‬ containing x.