General definitions of smoothing spline functions and basic conditions fulfilled by them are presented (Eqs. (5), (6) and (9)). The way of applying the theory to particular cases, namely polynomial smoothing spline functions of an odd order (Theorems 1, 2 and 3), is also shown. The construction of cubic spline free functions assuming that their first derivatives are given is described (Eqs. (22) and (23)). The equations of approximation error (Eqs. (24), (25) and (28)), of smoothing error (Eqs. (26) and (29)) and of global error (Eq. (27)) are given. The properties of these errors (Table 2, Figs, 3, 4) and the criterion of optimality of the parameter p resulting from the properties of the approximation error are shown (Example 2). The influence of the parameter p on the smoothing spline function (Figs. 1, 2 and 5) is examined in Examples 1-3. The analysis of the influence of random errors of data on the approximation error (Table 3) is given in Example 4. It occurs that the approximation error is less than the random error of data. In Example 5 we showed how the first derivative of the analysed function was estimated by the smoothing spline function.