Interpolating and smoothing biquadratic spline. (English) Zbl 0835.41016

Summary: The paper deals with the biquadratic splines and their use for the interpolation in two variables on the rectangular mesh. The possibilities are shown how to interpolate function values, values of the partial derivative or values of the mixed derivative. Further, the so-called smoothing biquadratic splines are defined and the algorithms for their computation are described. All of these biquadratic splines are derived by means of the tensor product of the linear spaces of the quadratic splines and their bases are given by the so-called fundamental splines.


41A15 Spline approximation
41A05 Interpolation in approximation theory
65D05 Numerical interpolation
65D07 Numerical computation using splines
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[1] J.H. Ahlberg, E.N. Nilson, J.L. Walsh: The Theory of Splines and their Applications. Academic Press, New York-London, 1967. · Zbl 0158.15901
[2] C. de Boor: Bicubic Spline Interpolation. J. Math. and Physics, 41 (1962), 212-218. · Zbl 0108.27103
[3] C. de Boor: A Practical Guide to Splines. Springer Verlag, New York, 1978. · Zbl 0406.41003
[4] S. Ewald, H. Mühlig, B. Mulansky: Bivariate Interpolating and Smoothing Tensor Product Splines. Proceeding ISAM, Berlin, 1989, pp. 59-68. · Zbl 0697.41004
[5] A. Imamov: About some Properties of Multivariate Splines. Vyčislitelnye sistemy (Novosibirsk) 65 (1975), 68-73.
[6] J. Kobza: An Algorithm for Biquadratic Spline. Appl. Math. 32 (1987), no. 5, 401-413. · Zbl 0635.65006
[7] J. Kobza: Quadratic Splines Smoothing the First Derivatives. Appl. Math. 37 (1992), no. 2, 149-156. · Zbl 0757.65006
[8] J. Kobza, R. Kučera: Fundamental Quadratic Splines and Applications. Acta UPO 32 (1993), 81-98. · Zbl 0803.41011
[9] J. Kobza, D. Zápalka: Natural and Smoothing Quadratic Spline. Appl. Math. 36 (1991), no. 3, 187-204. · Zbl 0731.65006
[10] G. Nürnberger: Approximation by Spline Function. Springer Verlag, New York, 1989. · Zbl 0692.41017
[11] J.S. Zavjalov, B.I. Kvasov, V.L. Miroshnichenko: Methods of Spline Functions. Nauka, Moscow, 1980. · Zbl 0524.65007
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