During the past decade, there have been an increasing number of papers devoted to the use of polynomial splines in signal processing [1]-[4]. The interest in these techniques grew after it was shown that most classical spline- fitting problems on a uniform grid (interpolation, least squares, and smoothing splines) could be solved efficiently using recursive digital filtering techniques. These spline-based algorithms have been found to be quite advantageous for image processing and medical imaging, especially in the context of high-quality interpolation, where it has been demonstrated that they yield the best cost-quality tradeoff among all linear techniques [5]-[8]. Polynomial splines have also been shown to play a fundamental role in wavelet theory [9].

Although there are a few applications of polynomial splines in continuous-time signal processing, splines have apparently had less impact in this area. Part of the reason may be that (piecewise) polynomials do only appear marginally in basic systems theory. The most prominent functions in continuous-time signal-and-systems theory are the exponentials, which correspond to the modes of differential systems (analog filters and circuits). Having made this observation and motivated by the search for a unification between the continuous and discrete-time approaches to signal processing, Unser [10] deals with the task of extending the previously mentioned formulation to the enlarged class of exponential splines. These splines, as their name suggests, are made up of exponential segments that are connected together in a smooth fashion. They form a natural extension of the polynomial splines and have been characterized mathematically in relatively general terms [11] [12].

The kind of splines that are the most appropriate for signal processing are the cardinal ones, which are defined on a uniform grid. Mathematically, this corresponds to the simplest possible setup, which goes back to the pioneering work of Schoenberg on polynomial splines in 1946 [13]. In his early paper, it was shown that every cardinal polynomial spline has a unique and stable representation in terms of B-spline expansion. For cardinal exponential spline which is general concept of polynomial spline, Unser showed that if one excludes the pathological case of improperly spaced imaginary roots, then every cardinal exponential spline with suitable parameter has a unique and stable representation in terms of its B-spline expansion [10].

We present several methods to implement exponential B-splines. The paper is organized as follows. In Section 2, we begin with abrief introduction of exponential B-splines and notations needed throughout the paper. In Section 3, the methods will be presented. The conclusion is given in Section 4.

In general, E-splines with parameter with is compactly supported and N-2 times differentiable so that it has a convergent Poisson summation formula as

.

From the Poisson summation formula of E-splines, we take a finite number of terms of the summation as an approximation of E-splines and find its truncation error bound as follows:

Theorem 3.1. Define ()-term approximation of E-splines as

,

and let. Then we have

Proof. Since

and

we get

The proof completes by.

In this paper, we present the method to implement exponential B-splines by its Poisson summation formula. We achieve an explicit formula on the truncation error bound for exponential B-spline. As the future work, one can generalize de Boor’s order recursion for the calculation of B-splines [15] into that for E-splines.

The author thanks Prof. Michael Unser for useful comments and suggestions during the preparation of the manuscript.

Sinuk Kang, (2016) On the Implementation of Exponential B-Splines by Poisson Summation Formula. Journal of Applied Mathematics and Physics,04,637-640. doi: 10.4236/jamp.2016.44072