= Spline quasi-interpolation
{title2=$Qf=\sum_i\lambda_i(f)N_i$}
A <spline> quasi-interpolant synthesizes a <B-spline> expansion from local bounded <linear functionals>, instead of solving a global interpolation system. Suppose each functional is bounded by $c_k\|f\|_{C[t_i,t_{i+k}]}$ and the operator reproduces every <spline>. On a knot cell $[t_j,t_{j+1}]$, only indices $j+1-k\le i\le j$ contribute. Their <supports> lie in $[t_{j+1-k},t_{j+k}]$, and <subpartition of unity for B-splines> proves local <operator norm> at most $c_k$. Reproduction of degree-$k-1$ <polynomials> and a local <Taylor theorem> remainder then give error at most $(1+c_k)(2k-1)^k h^k\|f^{(k)}\|_\infty/k!$, on the basic knot domain with the usual completed boundary <basis>.
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