Analytical formulas for polynomial coefficients in radial basis function interpolation

Radial basis functions (RBF) are used in many areas, including interpolation and approximation, solution of partial differential equations, neural networks, and machine learning. RBFs are based on the sum of weighted kernel functions. Additional orthogonal polynomials are added for robustness, numerical stability, and computational efficiency improvement.This contribution gives a new analytical formula specifying values of the polynomial coefficients used in RBF interpolation. The zerodegree polynomial coefficient is related to the sigmoid function used in RBF-neural networks (RBF-NN).Unlike prior works where polynomial augmentation is only used to guarantee solvability, this paper provides explicit closed-form formulae for polynomial coefficients (with special focus on the zerodegree case). This new analytical treatment clarifies their role as global bias terms in both interpolation and RBF neural networks. Expected applicability is in data interpolation and approximation, RBF-neural networks, scientific computing and PDE solutions, geostatistics & spatial interpolation, machine learning, and data fitting and signal processing.