Abstract
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.
Highlights
During the past decade, there have been an increasing number of papers devoted to the use of polynomial splines in signal processing [1]-[4]
This particular framework allows for important simplifications and that it is ideally suited for a signal processing formulation
We introduce the exponential B-spline βα representation theorem, which is a generalization of Schoenberg’s classical result [13] for cardinal polynomial splines and which shows the implementation of E-splines is enough to get for a signal processing using exponential spline
Summary
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 splinefitting problems on a uniform grid (interpolation, least squares, and smoothing splines) could be solved efficiently using recursive digital filtering techniques. 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.
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