Open Problem: Existence of Hermite Interpolatory Subdivision Schemes with Arbitrary Large Smoothnesses

Abstract

In the case of “Lagrange” interpolatory subdivision schemes, it is known that the family of Deslauriers-Dubuc schemes [2] furnishes a sequence of schemes with increasing supports and increasing smoothnesses [1]. To be more specific, a member of this family is an interpolatory subdivision scheme L _N of the form f ^k+1=L _N f ^k, given by the rules

$$f_{2\alpha }^{k + 1} = f_\alpha ^k, \alpha \in \mathbb{Z}, k \geqslant 0$$

((1))

$$f_{2\alpha + 1}^{k + 1} = {P_{2N,\alpha }}(\tfrac{1}{2}), \alpha \in \mathbb{Z}, k \geqslant 0$$

((2))

, where P_2N,α is a polynomial of degree 2N − 1, satisfying the Lagrange interpolation conditions:

$${P_{2N,\alpha }}(j) = f_{\alpha + j}^k, j = N + 1, - N + 2,..., - 1,0,1,...,N$$

((3))

.