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Generating functions for B-splines with knots in geometric or affine progression. (English) Zbl 1316.65020
The authors obtain generating functions for B-splines. Since there is no structure on the knots when they are completely arbitrary, any interesting property about the corresponding B-splines cannot be derived. Here explicit formulas for the generating functions of B-splines with knots sequences \(\{x_k\}\) in affine progression \(x_{k+1}=qx_k+h\) are given, with \(q\) and \(h\) fixed parameters, and in the particular case (\(h=0\)) of geometric progression they are considered in terms of \(q\)-exponential functions.
To find an explicit formula for the generating functions by the de Boor recurrence over knots in affine progression and the definition of \(q\)-derivative, a partial differential equation is constructed, whose solution is the generating function for B-splines in affine progression.
Such generating functions are then used to obtain both known and new identities, as a generalization of the Schoenberg identity and an explicit expression for the moments of these B-splines.
Finally, this discussion includes two interesting special cases, i.e., uniform B-splines with knots at the integers (\(q=h=1\)) and nonuniform B-splines with knots at the \(q\)-integers (\(q\not=1\), \(h=1\)).

MSC:
65D07 Numerical computation using splines
05A30 \(q\)-calculus and related topics
05A15 Exact enumeration problems, generating functions
41A15 Spline approximation
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