Noname manuscript No. (will be inserted by the editor) From approximating to interpolatory non-stationary subdivision schemes with the same reproduction properties Costanza Conti · Luca Gemignani · Lucia Romani Received: date / Accepted: date 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. It is shown that the interpolatory schemes are (mostly) capable of generating the same functional space as the approximating one. Moreover, the interplay between structured matri- ces and polynomials provides an effective tool for designing efficient numeric and/or numeric-symbolic methods for their construction and analysis. Keywords Subdivision schemes · Structured matrices · Polynomials Mathematics Subject Classification (2000) MSC 65F05 · MSC 65D05 1 Introduction This paper is the generalization of our recent work [2] to the non-stationary situation. In fact, many important subdivision schemes are of non-stationary nature like those able to reproduce conic sections, spirals or classical trigonometric curves which are important analytical shapes in geometric modeling. In particular we discuss how it is possible to move from a non-stationary approximating subdivision scheme to a non- stationary interpolatory one. C. Conti Dipartimento di Energetica “Sergio Stecco”, Universit`a di Firenze, Via Lombroso 6/17, 50134 Firenze, Italy E-mail: costanza.conti@unifi.it L. Gemignani Dipartimento di Matematica, Universit`a di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy E-mail: gemignan@dm.unipi.it L. Romani Dipartimento di Matematica e Applicazioni, Universit`a di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano, Italy E-mail: lucia.romani@unimib.it