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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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##### References:
 [1] Dikoussar, N.D., Math. model., 3, 10, 50, (1991), [in Russian] [2] Dikoussar, N.D., Comput. phys. commun., 79, 39, (1994) [3] Duda, R.; Hart, P., Pattern classification and scene analysis, (1973), Wiley New York · Zbl 0277.68056 [4] () [5] Klein, F., Vorlesungen über Höhere geometrie, (1926), Springer Berlin · JFM 52.0624.09 [6] Lanczos, C., Applied analysis, (1956), Prentice-Hall Englewood Cliffs, Verlag · Zbl 0111.12403 [7] Forsite, G.E.; Malcom, M.A.; Moler, C.B., Computer methods for mathematical computations, (1977), Prentice-Hall Englewood Cliffs, Verlag, H. Wind, Function Parametrization, CERN, 72-21 [8] Ahlberg, J.H.; Nilson, E.N.; Walsh, J.L., The theory of splines and their applications, (1967), Academic Press New York · Zbl 0158.15901 [9] Zavjalov, Y.S.; Kvasov, B.I.; Miroshnichenko, V.L.; Stechkin, S.B.; Subbotin, Y.N., Splines in computer mathematics, (1976), Moscow Nauka, [in Russian] [10] Popov, B.A., An uniform approximation by splines, (1984), Kiev Naukova Dumka, [in Russian] [11] Kalitkin, N.N.; Kusmina, L.V., Math. model., 6, 4, 77, (1994), [in Russian] [12] Roy, D.; Bhattacharya, R.; Bhowmick, S., Comput. phys. commun., 78, 29, (1993) [13] Nabhan, Tarek M.; Zamaya, Albert Y., Neural networks, 7, 1, 88, (1994)
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