On the (bi)infinite case of Shadrin’s theorem
نویسنده
چکیده
Some loose ends in Shadrin’s remarkable paper are tied up. Shadrin’s theorem [S] settles a problem posed first in [B73] in the following setting. For k ∈ N, let t := (ti : i ∈ Z) be nondecreasing with ti < ti+k, all i, and set a := inf i ti, sup i ti =: b. For each i, let Nik(x) := (ti+k − ti)∆(ti, . . . , ti+k)(x− ·) k−1 + = (∆(ti+1, . . . , ti+k)− ∆(ti, . . . , ti+k−1))(x − ·) k−1 + be the ith L∞-normalized B-spline of order k for the knot sequence t. For an arbitrary coefficient sequence c = (ci), the biinfinite sum ∑ i Nikci makes sense pointwise, i.e.,
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