On smoothing of data using Sobolev polynomials

Data smoothing is a method that involves finding a sequence of values that exhibits the trend of a given set of data. This technique has useful applications in dealing with time series data with underlying fluctuations or seasonality and is commonly carried out by solving a minimization problem with a discrete solution that takes into account data fidelity and smoothness. In this paper, we propose a method to obtain the smooth approximation of data by solving a minimization problem in a function space. The existence of the unique minimizer is shown. Using polynomial basis functions, the problem is projected to a finite dimension. Unlike the standard discrete approach, the complexity of our method does not depend on the number of data points. Since the calculated smooth data is represented by a polynomial, additional information about the behavior of the data, such as rate of change, extreme values, concavity, etc., can be drawn. Furthermore, interpolation and extrapolation are straightforward. We demonstrate our proposed method in obtaining smooth mortality rates for the Philippines, analyzing the underlying trend in COVID-19 datasets, and handling incomplete and high-frequency data.