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- W2030807589 abstract "Given an index set X , a collection3of subsets of X (all of the same cardinality), and a collection { l x } x ∈ X of commuting linear maps on some linear space, the family of linear operators whose joint kernel K = K (4) is sought consists of all l A ≔∏ a ∈ A l a with A any subset of X which intersects every B ∈5. The goal is to establish conditions, on6and l , which ensure that dim K (7)=∑ B ∈8 dim K ({ B }), or, at least, one or the other of the two inequalities contained 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 that9consists 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 on10and l . These conditions involve the solvability of certain linear systems of the form l b ?=φ b , b ∈ B , with B ∈11, and the existence of “placeable” elements of X , i.e., of x ∈ X for which every B ∈12not containing x has all but one element in common with some B ′∈13containing x ." @default.
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- W2030807589 date "1996-09-01" @default.
- W2030807589 modified "2023-09-29" @default.
- W2030807589 title "On Ascertaining Inductively the Dimension of the Joint Kernel of Certain Commuting Linear Operators" @default.
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- W2030807589 doi "https://doi.org/10.1006/aama.1996.0011" @default.
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