In this paper we consider a step function characterized by an arbitrary sequence of real-valued scalars and approximate it with a matching pursuit (MP) algorithm. We utilize a waveform dictionary with rectangular window functions as part of this algorithm. We show that the waveform dictionary is not necessary when all of the scalars are either non positive or non negative and the parameters of a wavelet dictionary on an integer lattice yields a closed-form solution for the initial optimization problem as part of the MP. Additionally, for any real-valued scalar sequence, we provide a solution with a related wavelet dictionary at each iteration of the algorithm. This allows for practical calculation of the approximating function, which we use to provide examples on simulated and real univariate time series data that display discontinuities in its underlying structure where the step function can be thought of as a sample from a signal of interest.

在本文中，我们考虑以任意实值标量序列为特征的阶跃函数，并使用匹配追踪 (MP) 算法对其进行近似。我们利用具有矩形窗函数的波形字典作为该算法的一部分。我们表明，当所有标量都是非正或非负时，波形字典不是必需的，并且整数格上的小波字典的参数为作为 MP 的一部分的初始优化问题产生了一个封闭形式的解。此外，对于任何实值标量序列，我们在算法的每次迭代中都提供了一个带有相关小波字典的解决方案。这允许对近似函数进行实际计算，