This work is one of a series of papers that is devoted to the further investigation of polynomial and trigonometric splines of the third order approximation. Polynomial basis splines are better known and therefore more commonly used. However, the use of trigonometric basis splines often provides a smaller approximation error. Here we discuss the interval estimation with the polynomial or trigonometric basis splines which are useful for the approximation of functions with one or two variables. For each grid interval we construct the approximation separately. We construct the interval estimation of approximation also separately on the each grid interval. The one-dimensional polynomial or trigonometric basis splines of the third order approximation are constructed when the values of the function are known in each point of interpolation. Numerical examples are represented.