Dual subdivision and interpolation
Luca Gemignani  1  , Lucia Romani  2@  , Alberto Viscardi  3  
1 : Università di Pisa
2 : Alma Mater Studiorum - Universitá di Bologna
3 : universitá di Torino

Subdivision schemes are well-established tools for the construction of curves and surfaces. Given an initial set of points attached to an initial grid, a subdivision scheme recursively refines the current grid and computes new data to be attached to it, in order to obtain a smooth limit. This talk concerns dual subdivision schemes working on one-dimensional grids and investigates their capability of generating limit curves that interpolate the initial data. Recalling the results in [2], we first show that, differently from primal schemes, their dual counterparts do not satisfy the step-wise interpolation property and are not defined via refinement rules that at each stage of the iteration leave the previous set of points unchanged. Then, after reviewing the work completed in [3], we prove that a whole family of dual interpolating schemes of arity larger than 2 can indeed be constructed and, as recently discussed in [1], its construction can be conveniently reduced to the solution of a certain Bezout-like polynomial equation. Moreover, under some suitable assumptions, it is also possible to study a priori the existence of a dual interpolating scheme of a certain arity, that fulfills specific requests about polynomial reproduction, support size and regularity.

[1] L.Gemignani, L.Romani, A.Viscardi. Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes. Adv. Comput. Math., 48(1), 4 2022.
[2] L. Romani. Interpolating m-refinable functions with compact support: the second generation classAppl. Math. Comput., 361:735-746, 2019.
[3] L. Romani, A. ViscardiDual univariate interpolatory subdivision of every arity: algebraic characterization and construction. J. Math. Anal. Appl., 484(1), 123713, 2020.

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