Sharp Bounds for the Lebesgue Constant in Quadratic Nodal Spline Interpolation

Abstract

It is known that, in the construction of polynomial spline approximation operators, the two properties of locality and interpolation at all of the knots are incompatible for quadratic and higher order splines. In previous work, the authors employed the concept of additional (or secondary) knots to explicitly construct a nodal spline approximation operator W which possesses, for arbitrary order m, the properties of locality, interpolation at particular (primary) knots, as well as exactness on the polynomial class of order m. In addition, estimates were obtained which implied a rather crude upper bound, depending only on m and the local primary mesh ratio, for the Lebesgue constant ∥W∥. The principal aim of this paper is to explicitly calculate, in the quadratic case m = 3, sharp upper and lower bounds for ∥W∥. The exact value ∥W∥ = 1.25 in the case of equidistant primary knots is then immediately derivable. Some implications of the results, as well as an application in quadrature, are pointed out and briefly discussed.