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 + ω ) ≤ b ψk(ω) ≤ Bk(1 + ω ), 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.