Histopolating splines

Abstract

Given an integrable function f, we are concerned with the construction of a spline H n (f ) of degree n with prescribed knots t = (t j ) j ∈Z that satisfies the histopolation conditionsfor some fixed s n+1 ∈ N. Additionally, the resulting spline operator should be local and reproduce all polynomials of degree n. Our approach of generating such a histospline is based on a local spline interpolation operator that is exact for all polynomials of degree n.

Algorithm to Construct Integro Splines

“…Interpolation of mean values or area-matching properties is often referred to as histospline or histopolation, which has been studied in the literature (see [3,13] and the references therein). The convergence properties of the histospline density estimate were studied by Wahba [14], where the reader can also find several references to earlier results on this topic.…”

Cubic Spline Histopolation*

“…The histopolation with splines is studied in many papers under different names like area matching interpolation [2,3,6], interpolation in the mean [4,6,20], interpolation of mean values [11], histospline [20]. The spline histopolation on biinfinite knot sequence is treated in [20]. There are several papers by quartic spline interpolation, e.g., [11,12,13] but the interpolation problem which is equivalent to the histopolation with cubic splines is not treated in them in such extent as we do in current paper.…”