A function of the form \(\sum^n_{i= 0} c_ip_i(x) \varphi(x,\vec a)\) is called a generalized interpolation polynomial with a basic function \(\varphi(x,\vec a)\) and node coefficients \(p_i(x)\in C[a,b]\). The authors show that the generalized interpolation polynomials allow one to solve a number of problems in nonlinear approximation theory; e.g., to restore an unknown function \(f\) one can use not only a table of values, but also some properties of \(f\) such as smoothness, asymptotic behaviour near some points, etc.