Bounded local spline quasi-interpolants

Fix a bounded interval $[a,b]$, a degree $p\in {\mathbb{N}}_0$, an exponent $1\le q\le \mathrm{\infty }$, an integer $0\le l\le p$, and a nonempty family $\mathcal{T}$ of clamped degree-$p$ knot meshes on $[a,b]$. A mesh-uniform bounded local quasi-interpolant is a family of real-linear maps $Q_{\xi }:C([a,b];\mathbb{R})\to S_{p,\xi }$, one for each $\xi \in \mathcal{T}$, for which there are common constants $M\in \mathbb{N}$ and $0\le C_Q<\mathrm{\infty }$ such that the following hold for every $\xi \in \mathcal{T}$ and every nonempty cell $I_m$ of its mesh: there is a neighborhood $J_{\xi ,m}\subseteq [a,b]$ containing $I_m$ and formed from at most $M$ adjacent knot cells; each point of $[a,b]$ belongs to at most $M$ of the neighborhoods $J_{\xi ,m}$; (i) if $f=g$ almost everywhere on $J_{\xi ,m}$, then $Q_{\xi }f=Q_{\xi }g$ almost everywhere on $I_m$; (ii) $Q_{\xi }v=v$ for every $v\in {\mathcal{P}}_l$; and (iii) $\mathrm{\parallel }{Q}_{\xi }f{\mathrm{\parallel }}_{L^q(I_m)}\le C_Q\mathrm{\parallel }f{\mathrm{\parallel }}_{L^q(J_{\xi ,m})}$. The constants $M$ and $C_Q$ are independent of $\xi$. The derivative and continuous-representative conventions for Sobolev inputs are those of Integer-order Sobolev spaces and their norms and One-dimensional $W^{1,p}$ functions have unique absolutely continuous representatives.

For a fixed mesh, the coefficient-functional formulation in Lyche–Manni–Speleers is one sufficient way to obtain these local stability and reproduction properties. Uniform constants in that construction give a family as defined here; this definition records the operator hypotheses used in the approximation proof.