分野別セミナー

This paper develops a framework to approximate functions when the data are sparse but a slow-moving pattern allows for a useful fit. This is done by investigating the properties of Lipschitz functions and extending the concept to wiggly functions by allowing a deviation. Deterministic bounds are found for the approximation error of various methods in terms of the Lipschitz constant and the deviation. Moreover this paper presents an optimal method which outperforms other methods such as nearest neighbor and linear interpolation. The developed methods can also use the extra assumption of periodicity to obtain better prediction error bounds. This work also finds optimal sampling times which minimize the approximation error.