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The idea of local smoothing and local approximation is so natural that it is not surprising it has appeared in many branches of science. Citing [145] we mention early works in statistics using local polynomials by the Italian meteorologist Schiaparelli (1866) and the Danish actuary Gram (1879) (famous for developing the Gram-Schmidt procedure for the orthogonalization of vectors). In the 1960s and 1970s the idea became the subject of intensive theoretical study and applications, in statistics by Nadaraya [158], Watson [230], Parzen [175], and Stone [215], and in engineering sciences by Brown [19], Savitzky and Golay [196], Petersen [179], Katkovnik [98], [99], [101], and Cleveland [29]. The local polynomial approximation as a tool appears in different modifications and under different names, such as moving (sliding, windowed) least square, Savitzky-Golay filter, reproducing kernels, and moment filters. We prefer the term LPA with a reference to publications on nonparametric estimation in mathematical statistics where the advanced development of this technique can be seen. In this chapter the discrete local approximation is presented in a general multivariate form. In the introductory section (Sec. 2.1), we discuss an observation model and multi-index notation for multivariate data, signals, and estimators. Section 2.2 starts with the basic ideas of the LPA presented initially for 2D signals typical for image processing. The window function, the order of the LPA model, and the scaling of the estimates are considered in detail. Further, the LPA is presented for the general multivariate case with estimates for smoothing and differentiation. Estimators for the derivatives of arbitrary orders are derived. Examples of 1D smoothers demonstrate that the scale (window size) parameter selection is of importance. The LPA estimates are given in kernel form in Sec. 2.3. The polynomial smoothness of the kernels and their properties are characterized by the vanishing moment conditions. The LPA can be treated as a design method for linear smoothing and differentiating filters. Links of the LPA with the nonparametric regression concepts are clarified in Sec. 2.4, and the LPA for interpolation is discussed briefly in Sec. 2.5.
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