On the Chebyshev norm of polynomial \(B\)-splines
The authors investigate polynomial \(B\)-splines of given order \(m\) and with knots of arbitrary multiplicity with respect to their Chebyshev norm. Let \({\mathcal B}_m\) be the set of all \(B\)-splines of order \(m\) with the normalization \(x_0= 0\) and \(x_k=1\). Let \[ \lambda_m:= \sup\{ |B_m |\mid B_m\in {\mathcal B}_m\} \qquad \text{and} \qquad \mu_m:= \inf \{|B_m |\in {\mathcal B}_m\}. \] A \(B\)- spline \(B_m\in {\mathcal B}_m\) is called maximal (minimal) if its norm is equal to \(\lambda_m\) \((\mu_m)\). First they establish some results on \(B\)-splines with a small number of knots, and a contour integral representation for \(B\)-splines and their derivatives. Then they present a complete characterization of those \(B\)-splines with maximal and those with minimal norm and compute these norms explicitly. They demonstrate that the sequence \(\{ \mu_m\}\) satisfies the asymptotic relation \[ \mu_m= \sqrt { \Biggl( {2 \over {\mu m}} \Biggr)} \Biggl( 1+ {1\over {4m}}+ O(m^{-2}) \Biggr) \qquad \text{for }m\to \infty, \] and the sequence of norms of the equidistant \(B\)-splines \[ \beta_m= \sqrt {\Biggl( {6\over {\pi m}} \Biggr)} \Biggl( 1- {3\over {2cm}}+ O(m^{-2}) \Biggr) \qquad \text{for }m\to \infty. \] They further study the problem of location and the behavior of the \(B\)-splines maximum, when a new knot is inserted or if one knot is moved. Finally they analyse those types of knot distributions for which the norms of the corresponding \(B\)-splines converge to zero as \(m\to \infty\). In fact they obtain necessary and sufficient conditions, under which the \(B\)-spline norms tend to zero as \(m\) tends to infinity.
scientific article; zbMATH DE number 782146