# Dual Univariate Interpolatory Subdivision of Every Arity: Algebraic Characterization and Construction

## TL;DR

This paper introduces a new class of dual univariate interpolatory subdivision schemes with algebraic characterization for all arities, offering improved properties over classical primal schemes.

## Contribution

It provides a complete algebraic characterization and a construction strategy for dual univariate interpolatory subdivision schemes of every arity.

## Key findings

- Schemes have masks with an even number of elements
- Examples show superior regularity and support length
- Applicable to a wide range of arities

## Abstract

A new class of univariate stationary interpolatory subdivision schemes of dual type is presented. As opposed to classical primal interpolatory schemes, these new schemes have masks with an even number of elements and are not step-wise interpolants. A complete algebraic characterization, which covers every arity, is given in terms of identities of trigonometric polynomials associated to the schemes. This characterization is based on a necessary condition for refinable functions to have prescribed values at the nodes of a uniform lattice, as a consequence of the Poisson summation formula. A strategy for the construction is then showed, alongside meaningful examples for applications that have comparable or even superior properties, in terms of regularity, length of the support and/or polynomial reproduction, with respect to the primal counterparts.