Functional precision set of a subdivision scheme ($\mathcal{P}=\{f:{\mathcal{R}}_h(f{|}_{h\mathbb{Z}})=f\}$)

The functional precision set of a subdivision scheme consists of functions exactly reproduced by refining their uniform samples with the scheme's consistent parameter placement. Polynomial precision usually specifies the reproduced polynomial space. If exactness is required for every grid spacing and shift, the binary midpoint-interpolation scheme has precision exactly the affine functions. At one fixed grid, its entire reconstruction range is the space of continuous piecewise linear functions on that grid; this larger range should not be confused with polynomial precision or arbitrary-grid reproduction.