High order smoothness of non-linear Lane-Riesenfeld algorithms in the functional setting

We investigate some variants of the linear Lane-Riesenfeld algorithm in the functional setting, generated by replacing the standard binary arithmetic averages between real numbers by non-linear, binary, symmetric averages between real numbers. For certain classes of non-linear averages we show that such generalized r-th order Lane-Riesenfeld algorithms generate limit functions with the same smoothness as the linear r-th order Lane-Riesenfeld algorithm. The analysis is based on the observation that a non-linear subdivision scheme can be regarded as a non-uniform linear scheme with mask coefficients depending on the initial data, and on ideas from a recent investigation of smoothly varying non-uniform linear subdivision schemes (Dyn et al., 2014). Our main result is motivated by an example in Duchamp et al. (2016). It extends a result in Duchamp et al. (2018), derived for subdivision schemes on smooth manifolds, and applied to the non-linear Lane-Riesenfeld algorithm in the functional setting, by allowing different non-linear averages at different locations in a smoothing step.