Function parametrization by using 4-point transforms Original Research Article

A continuous parametrization of the smooth curve f(x) = f(x; R) is suggested on a basis of four-point transformations. Coordinates of three reference points of the curve are chosen as parameters R. This approach allows to derive a number of advantages in function approximation and fitting of empiric data. The transformations have made possible to derive a new class of polynomials (monosplines) having the better approximation quality than monomials xn. A behaviour of an error of the approximation has an uniform character. A three-point model of the cubic spline (TPS) is proposed. The model allows to reduce a number of unknown parameters in twice and to obtain an advantage in a computing aspect. The new approach to the function approximation and fitting are shown on a number of examples. The proposed approach gives a new mathematical tool and a new possibility in both practical applications and theoretical research of numerical and computational methods.