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Polynomial splines have played an important role in image processing, medical imaging and wavelet theory. Exponential splines which are of more general concept have been recently investigated.We focus on cardinal exponential splines and develop a method to implement the exponential B-splines which form a Riesz basis of the space of cardinal exponential splines with finite energy.

During the past decade, there have been an increasing number of papers devoted to the use of polynomial splines in signal processing [

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 [

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 [

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.

Vectors are marked with an arrow and are used to represent N-tuples, i.e.,

The Fourier transform of

otherwise, it is defined in the distributional sense. The Laplace transform of a causal (possibly exponentially increasing) function

The one-sided power function is

Let us consider the generic differential operator of order N

with constant coefficients

We will therefore use the notation

Definition 2.1. An exponential spline with parameter

where the sequence

The cardinal exponential splines correspond to the specialized case where the knots are at the integer, i.e.,

We now introduce the exponential B-spline

Theorem 2.2. [

Thus, if one excludes the pathological cases of improperly spaced imaginary roots first identified by Ron [

The E-splines are localized, that is compactly supported and shortest possible, versions of the Green functions that generate the exponential splines. The way in which such E-splines are constructed is especially easy to understand in the first-order case. One takes the green function

Note that this first-order E-spline is supported in [0, 1), irrespective of

The higher order E-splines are obtained by successive convolution of lower order ones:

which is a process that is justified by the convolution relation of the corresponding Green functions.

In general, E-splines with parameter

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 (

and let

Proof. Since

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 [

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