Atomic Exponential Basis Function Eup(x, omega) - Development and Application

This paper presents exponential Atomic Basis Functions (ABF), which are called Eup (x, omega)). These functions are infinitely differentiable finite functions that unlike algebraic up(x) basis functions, have an unspecified parameter- frequency omega). Numerical experiments show that this class of atomic functions has good approximation properties, especially in the case of large gradients (Gibbs phenomenon). In this work, for the first time, the properties of exponential ABF are thoroughly investigated and the expression for calculating the value of the basis function at an arbitrary point of the domain is given in a form suitable for implementation in numerical analysis. Application of these basis functions is shown in the function approximation example. The procedure for determining the best frequencies, which gives the smallest approximation error in terms of the least squares method, is presented.