Minimal degree univariate piecewise polynomials with prescribed Sobolev regularity

Compactly Supported, Piecewise Polyharmonic Radial Functions with Prescribed Regularity

“…, ±k. It is shown in [2] that ψ k has regularity (1, k), which is regularity (1, k, 0) in the language of the previous section. In this section, we first prove that ψ k has regularity (1, k, 1), and then we define and discuss the regularity of the families {ψ d,k } and {ψ d,k,m }.…”

“…, a k are as given in (3.3). In [2], (3.2) was encountered with H(z) = cos z, and it was shown that 1 + k j=1 a j cos(jt) = α k (1 − cos t) k , where α k > 0 is defined by…”

“…It is shown in [2] that F 1 ψ k can be written in the form (1 − cos t) dt = π. Proceeding by induction, assume that (i) holds for k and consider k + 1.…”

“…Since ψ k is positive on (0, k) and 0 elsewhere, and with Theorem 7.3 in view, it follows that For k ∈ N, the function ψ k is piecewise in Z 1 and belongs to C 2k−1 (0, ∞) (see [2]). It follows from Corollary 6.2 that for j = 1, 2, .…”

“…Other noteworthy constructions are those of Buhmann [4], who constructed "single-piece" piecewise functions of the form φ • ρ d , where φ = qχ [0,1] with q analytic on (0, 1], as well as several families constructed by Gneiting (see [8] and the references therein). Recently, Al-Rashdan and the author (see [2]) showed that the B-spline ψ k , having simple knots at {±1, ±2, . .…”