BSplineKerne

BSplineKerne is a kernel function derived from B-spline basis functions. It is used for smoothing, interpolation, and as a building block in kernel-based methods. In one dimension, a B-spline of order n is a piecewise polynomial with compact support, obtained by convolving a simple starting function with itself n times. The corresponding kernel K_n is typically taken to be the B-spline function B_n, sometimes centered and normalized to integrate to one. In higher dimensions, a common choice is a separable kernel K_n(x) = product over coordinates of B_n(x_i), which makes convolution efficient and yields predictable smoothing along each axis.

Key properties of B-spline kernels include non-negativity, symmetry about the center, and compact support whose length grows with the order. The smoothness increases with the order: a B-spline of order n is C^{n-2} continuous for n ≥ 2. The kernel is refinable, supporting multi-resolution representations, and, with proper normalization, forms a partition of unity. These traits make B-spline kernels well suited for local smoothing and interpolation tasks.

Common applications span image and signal processing, where cubic B-spline kernels (order four) are widely used for resampling, anti-aliasing, and reconstruction, as well as computer graphics and numerical analysis. In machine learning, B-spline kernels appear as a family of kernel functions for nonparametric regression and some kernel-based classifiers, offering a tunable balance between locality and smoothness. BSplineKerne thus represents a versatile, mathematically well-characterized class of smoothing kernels with wide practical use.