Optimal interpolation formulas in the space \(W_2^{(m,m-1)}\). (English) Zbl 1457.41001
Summary: In the present paper we investigate the problem of construction of the optimal interpolation formulas in the space \(W_2^{(m,m-1)}(0,1)\). We find the norm of the error functional which gives the upper bound for the error of the interpolation formulas in the space \(W_2^{(m,m-1)}(0,1)\). Further we get the system of linear equations for coefficients of the optimal interpolation formulas. Using the discrete analogue of the differential operator \(\frac{\,\text{d}^{2m}}{\,\text{d}x^{2m}}-\frac{\,\text{d}^{2m-2}}{\,\text{d}x^{2m-2}}\) and its properties we find explicit formulas for the coefficients of the optimal interpolation formulas. Finally, we give some numerical results which the confirm theoretical results of the paper.
MSC:
| 41A05 | Interpolation in approximation theory |
| 65D30 | Numerical integration |
| 65D32 | Numerical quadrature and cubature formulas |
References:
| [1] | Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The Theory of Splines and their Applications, Mathematics in Science and Engineering. Academic Press, New York (1967) · Zbl 0158.15901 |
| [2] | Arcangeli, R., Lopez de Silanes, M.C., Torrens, J.J.: Multidimensional Minimizing Splines, p. 261. Kluwer Academic publishers, Boston (2004) · Zbl 1064.41001 · doi:10.1007/b130045 |
| [3] | Attea, M.: Hilbertian kernels and spline functions. In: Brezinski, C., Wuytack, L. (eds.) Studies in Computational Matematics 4. North-Holland, Amsterdam (1992) · Zbl 0767.41015 |
| [4] | Avezova, N.R., Samiev, K.A., Hayotov, A.R., Nazarov, I.M., Ergashev, ZZh, Samiev, M.O., Suleimanov, ShI: Modeling of the unsteady temperature conditions of solar greenhouses with a short-term water heat accumulator and its experimental testing. Appl. Solar Energy 46, 4-7 (2010) · doi:10.3103/S0003701X10010020 |
| [5] | Bezhaev, AYu., Vasilenko, V.A.: Variational Theory of Splines, p. 269. Springer, New York (2001) · Zbl 0979.41007 · doi:10.1007/978-1-4757-3428-7 |
| [6] | de Boor, C.: Best approximation properties of spline functions of odd degree. J. Math. Mech. 12, 747-749 (1963) · Zbl 0116.27601 |
| [7] | de Boor, C.: A Practical Guide to Splines. Springer, Berlin (1978) · Zbl 0406.41003 · doi:10.1007/978-1-4612-6333-3 |
| [8] | Hayotov, A.R., Babaev, S.S.: Calculation of the coefficients of optimal interpolation formulas in the space \[W_2^{(2,1)}(0,1)\] W2(2,1)(0,1). Uzb. Math. J. 3, 126-133 (2014). (in Russian) |
| [9] | Hayotov, A.R., Kholova, S.O., Mamatova, N.H.: An optimal interpolation formula and an interpolation spline minimizing the semi-norm in the space \[W_2^{(1,0)}(0,1)\] W2(1,0)(0,1). Uzb. Math. J. 2, 121-126 (2015). (in Russian) |
| [10] | Holladay, J.C.: Smoothest curve approximation. Math. Tables Aids Comput. 11, 223-243 (1957) · Zbl 0084.34904 · doi:10.2307/2001941 |
| [11] | Ignatev, M.I., Pevniy, A.B.: Natural Splines of Many Variables. Nauka, Leningrad (1991). (in Russian) · Zbl 0732.41012 |
| [12] | Laurent, P.-J.: Approximation and Optimization, p. 496. Mir, Moscow (1975). (in Russian) |
| [13] | Schoenberg, I.J.: On trigonometric spline interpolation. J. Math. Mech. 13, 795-825 (1964) · Zbl 0147.32104 |
| [14] | Schumaker, L.L.: Spline Functions: Basic Theory, p. 600. Cambridge University Press, Cambridge (2007) · Zbl 1123.41008 · doi:10.1017/CBO9780511618994 |
| [15] | Shadimetov, Kh.M., Hayotov, A.R.: Construction of the discrete analogue of the differential operator \[d^{2m}/dx^{2m}-d^{2m-2}/dx^{2m-2}\] d2m/dx2m-d2m-2/dx2m-2. Uzb. Math. J. 2, 85-95 (2004) |
| [16] | Shadimetov, Kh.M., Hayotov, A.R.: Properties of the discrete analogue of the differential operator \[d^{2m}/dx^{2m}-d^{2m-2}/dx^{2m-2}\] d2m/dx2m-d2m-2/dx2m-2. Uzb. Math. J. 4, 72-83 (2004) |
| [17] | Shadimetov, Kh.M., Hayotov, A.R.: Construction of interpolation splines minimizing semi-norm in \[W_2^{(m, m-1)}(0,1)\] W2(m,m-1)(0,1) space. BIT Numer. Math. 53, 545-563 (2013) · Zbl 1273.41007 |
| [18] | Shadimetov, Kh.M., Hayotov, A.R.: Optimal quadrature fromulas in the sense of Sard in \[W_2^{(m, m-1)}\] W2(m,m-1) space. Calcolo 51, 211-243 (2014) · Zbl 1312.65035 |
| [19] | Sobolev, S.L.: On interpolation of functions of \[n\] n variables. In: Selected Works of S.L. Sobolev, pp. 451-456. Springer, Berlin (2006) |
| [20] | Sobolev, S.L.: Formulas of mechanical cubature in \[n\] n-dimensional space, In: Selected Works of S.L. Sobolev, pp. 445-450, Springer, Berlin (2006) |
| [21] | Sobolev, S.L.: Introduction to the Theory of Cubature Formulas, p. 808. Nauka, Moscow (1974) |
| [22] | Sobolev, S.L., Vaskevich, V.L.: The Theory of Cubature Formulas. Kluwer Academic Publishers Group, Dordrecht (1997) · Zbl 0877.65009 · doi:10.1007/978-94-015-8913-0 |
| [23] | Sobolev, S.L.: The coefficients of optimal quadrature formulas, In: Selected Works of S.L.Sobolev, pp. 561-566, Springer, Berlin (2006) |
| [24] | Stechkin, S.B., Subbotin, YuN: Splines in Computational Mathematics, p. 248. Nauka, Moscow (1976). (in Russian) · Zbl 0462.65007 |
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