On approximations of the sixth order with the smooth polynomial and non-polynomial splines

This paper discusses twice continuously differentiable and three times continuously differentiable approximations with polynomial and non-polynomial splines. To construct the approximation, a polynomial and non-polynomial local basis of the second level and the sixth order approximation is constructed. We call the approximation a second level approximation because it uses the first and the second derivatives of the function. The non-polynomial approximation has the properties of polynomial and trigonometric functions. Here we have also constructed a non-polynomial interpolating spline which has the first, the second and the third continuous derivative. This approximation uses the values of the function at the nodes, the values of the first derivative of the function at the nodes and the values of the second derivative of the function at the ends of the interval [a, b]. The theorems of the approximations are given. Numerical examples are given.