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Shift-invariant kernels and estimates are introduced in Sec. 3.1, where the formulas for the kernels are derived. The smoothness of the shift-invariant kernels and the reproducing properties of the estimates are specified. The corresponding vanishing moment conditions are presented in Sec. 3.2. The frequency domain characterization of the LPA kernels and estimates are presented in Sec. 3.3. Using orthonormal polynomials allows us to obtain analytical formulas for smoothing and differentiating LPA kernels. Numerical 2D examples of LPA filters are discussed in Sec. 3.4 for uniform and Gaussian windows. In Sec. 3.5 the conventional differentiators are considered with connections to the general LPA approach.
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