On the Chebyshev Norm of Polynomial B-Splines

Abstract

Polynomial B-splines of given order m and with knots of arbitrary multiplicity are investigated with respect to their Chebyshev norm. We present a complete characterization of those B~splines with maximal and with minimal norm, compute these norms explicitly and study their behavior as m tends to infinity. Furthermore, the norm of the B-spline corresponding to the equidistant distribution of knots is studied. Finally, we analyse those types of knot distributions, for which the norms of the corresponding B-splines converge to zero as m .......• 00 .

A Reduction Method for Linear Functionals and Its Applications

“…Next, the case t = 1 is considerd in more detail. This result was also proven by Diethehn [5], and for B-splines by Meinardus, ter Morsche and Walz [8]. Next, a factorization of ^ is considered, which is based on the foUowing lemma.…”

“…In section 2, the definiteness criterion of Brass and Schmeisser [2] and their approach to error estimates are shortly reviewed. Section 3 contains the basic theory of this paper, i.e., relations between the kernels of R and those of R", which generalizes results of Meinardus, ter Morsche and Walz [8] on B-spIines, and also results of Diethelm [5]. In the following three sections, some applications are given: derivation of £-spUne identities in section 4, monotonicity of Compound quadrature formulae in section 5 (where the approach of Newman [9] to the monotonicity of Compound Newton-Cotes formulae is generalized), and some other possible applications and generalizations are indicated in section 6.…”

Interlacing property for B-splines

“…For the convenience of the reader, we include a proof which, unlike [8], does not involve the integral representation of divided differences.…”

“…Our proof of the monotonicity property for polynomial B-splines makes an extensive use of an elegant formula which was given by Meinardus et al [8,Theorem 5] and which was expressed in a slightly different way by Chakalov [5] as early as 1938 (see also [3,Formula (3.4.6)]). For the convenience of the reader, we include a proof which, unlike [8], does not involve the integral representation of divided differences.…”

Enhanced Convergence Estimates for Semi-Lagrangian Schemes Application to the Vlasov--Poisson Equation

“…This new error estimate is based on new estimates for the advection equation on a regular periodic grid for which we use sharp properties of semi-lagrangian schemes also equal to shifted Strang's stencil [14,8,9], see also [2,3,4,5]. We will make use of the connection of these numerical schemes with B-Splines techniques [11,13].…”