Abstract
Nonlinear approximation is widely used in signal processing. Real-life signals can be modeled as functions of bounded variation. Thus the variable knot of approximating function could be self- adaptively chosen by balancing the total variation of the target function. In this paper, we adopt continuous piecewise linear approximation instead of the existing piecewise constants approximation. The results of experiments show that this new method is superior to the old one.
Highlights
The fundamental problem of approximation theory is to resolve a possibly complicated function, called the target function, by simpler, easier to compute functions called the approximants [1] [2]
The approximating function is not restricted to come from spaces of piece wise polynomials with a fixed partition; rather, the partition was allowed to depend on the target function
In order to self-adaptively select the knots according to the target function, we can balance the total variation (TV) of this function [4]
Summary
The fundamental problem of approximation theory is to resolve a possibly complicated function, called the target function, by simpler, easier to compute functions called the approximants [1] [2]. The early methods utilized approximation from finite-dimensional linear spaces. In the beginning, these were typically spaces of polynomials, both algebraic and trigonometric. It was noted shortly thereafter that there were some advantages to be gained by not limiting the approximations to come from linear spaces [2]. The approximating function is not restricted to come from spaces of piece wise polynomials with a fixed partition; rather, the partition was allowed to depend on the target function. An important question in nonlinear approximation is how we should measure this smoothness in order to obtain definitive results
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