Minimal degree univariate piecewise polynomials with prescribed Sobolev regularity

For k∈{1,2,3,…}, we construct an even compactly supported piecewise polynomial ψk whose Fourier transform satisfies Ak(1+ω2)−k≤ψ̂k(ω)≤Bk(1+ω2)−k, ω∈R, for some constants Bk≥Ak>0. The degree of ψk is shown to be minimal, and is strictly less than that of Wendland's function ϕ1,k−1 when k>2. This shows that, for k>2, Wendland's piecewise polynomial ϕ1,k−1 is not of minimal degree if one places no restrictions on the number of pieces.