From approximating to interpolatory non-stationary subdivision schemes with the same generation properties

TL;DR

This paper presents a general method to derive interpolatory non-stationary subdivision schemes from non-interpolatory ones, ensuring they reproduce the same exponential polynomial space and demonstrating their effectiveness through computational examples.

Contribution

The authors extend previous work to remove assumptions on input symbols, providing a feasible strategy to generate interpolatory schemes with identical reproduction properties.

Findings

Interpolatory schemes reproduce the same exponential polynomial space as original schemes.

The method applies to symbols from shifted non-stationary affine combinations of exponential B-splines.

Proposed schemes exhibit smoothness and strong reproduction properties in computational examples.

Abstract

In this paper we describe a general, computationally feasible strategy to deduce a family of interpolatory non-stationary subdivision schemes from a symmetric non-stationary, non-interpolatory one satisfying quite mild assumptions. To achieve this result we extend our previous work [C.Conti, L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] to full generality by removing additional assumptions on the input symbols. For the so obtained interpolatory schemes we prove that they are capable of reproducing the same exponential polynomial space as the one generated by the original approximating scheme. Moreover, we specialize the computational methods for the case of symbols obtained by shifted non-stationary affine combinations of exponential B-splines, that are at the basis of most non-stationary subdivision schemes. In this case we find that the associated family of interpolatory symbols can be determined to satisfy a suitable set of generalized interpolating conditions at the set of the zeros (with reversed signs) of the input symbol. Finally, we discuss some computational examples by showing that the proposed approach can yield novel smooth non-stationary interpolatory subdivision schemes possessing very interesting reproduction properties.