Fractional generalized splines and signal processing

This paper presents the concept of fractional generalized splines, which is an extension of the idea of Unser's fractional splines. The first part of this paper describes a method for construction of fractional generalized splines through evaluating fractional finite differences. The main key to our approach is to provide an additional tuning parameter ‘α’ by using a generating function, which is the solution of the Laguerre's nth-order differential equation. The second part of the paper deals with characterization of these functions in both time and frequency domain and shows how to use these results for construction of wavelet bases in L2 for signal processing applications. This paper also present simulation results to reveal the suitability of the proposed basis functions for signal approximation.