Limits of level and parameter dependent subdivision schemes: a matrix approach
This work provides a theoretical tool for analyzing non-stationary subdivision schemes, which is relevant for researchers in approximation theory and geometric modeling.
The paper introduces a matrix approach using joint spectral radius to analyze convergence and Hölder regularity of subdivision schemes that depend on both level and parameters varying in a compact set. The method is shown to be efficient and improves upon stationary schemes in examples.
In this paper, we present a new matrix approach for the analysis of subdivision schemes whose non-stationarity is due to linear dependency on parameters whose values vary in a compact set. Indeed, we show how to check the convergence in $C^{\ell}(\RR^s)$ and determine the Hölder regularity of such level and parameter dependent schemes efficiently via the joint spectral radius approach. The efficiency of this method and the important role of the parameter dependency are demonstrated on several examples of subdivision schemes whose properties improve the properties of the corresponding stationary schemes. Moreover, we derive necessary criteria for a function to be generated by some level dependent scheme and, thus, expose the limitations of such schemes.