Normalized b-spline construction for smooth splines and quasi-interpolation

Abstract

In this paper, we introduce a general framework for constructing univariate spline spaces defined on refined non-uniform partitions with variable knot insertion. The proposed approach is developed for spline spaces characterized by a prescribed level of smoothness and a corresponding range of polynomial degrees. Within this setting, we establish suitable continuity conditions at the inserted knots and develop a fully local Hermite interpolation scheme. We then construct a normalized B-spline-type basis consisting of non-negative, compactly supported functions that form a partition of unity. By means of blossoming techniques, we derive adapted B-spline-type representations and design quasi-interpolants exhibiting superconvergence properties. The proposed operators provide high-order accuracy for approximation on non-uniform meshes, as confirmed by theoretical analysis and numerical experiments.