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Function parametrization by using 4-point transforms. (English) Zbl 0927.65009
Summary: A continuous parametrization of the smooth curve \(f(x)= f(x;{\mathcal R})\) is suggested on a basis of four-point transformations. Coordinates of three reference points of the curve are chosen as parameters \({\mathcal R}\). 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 \(\{x^n\}\). A behaviour of an error of the approximation has a uniform character. A three-point model of the cubic spline is proposed. The model allows to reduce the number of unknown parameters in half 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.

MSC:
65D10 Numerical smoothing, curve fitting
65D07 Numerical computation using splines
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