S.I. Novikov. On an interpolation problem with the smallest $L_2$-norm of the Laplace operator ... P. 143-153

The paper is devoted to an interpolation problem for finite sets of real numbers bounded in the Euclidean norm. The interpolation is by a class of smooth functions of two variables with the minimum $L_{2}$-norm of the Laplace operator $\Delta =\partial^{2 }/\partial x^{2}+\partial^{2 }/\partial y^{2}$ applied to the interpolating functions. It is proved that if $N\geq 3$ and all interpolation points $\{(x_{j},y_{j})\}_{j=1}^{N}$ do not lie on the same line, then the minimum value of the $L_{2}$-norm of the Laplace operator on interpolants from the class of smooth functions for interpolated data from the unit ball of the space $l_{2}^{N}$ is expressed in terms of the largest eigenvalue of the matrix of a certain quadratic form.

Keywords: interpolation, Laplace operator, thin plate splines

Received August 19, 2022

Revised September 1, 2022

Accepted September 5, 2022

Sergey Igorevich Novikov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: Sergey.Novikov@imm.uran.ru

REFERENCES

1.   Subbotin Yu.N. Functional interpolation in the mean with smallest n derivative. Proc. Steklov Inst. Math., 1967, vol. 88, pp. 31–63.

2.   Subbotin Yu.N., Novikov S.I., Shevaldin V.T. Extremal functional interpolation and splines. Trudy Inst. Mat. i Mekh. UrO RAN, 2018, vol. 24, no. 3, pp. 200–225 (in Russian). doi: 10.21538/0134-4889-2018-24-3-200-225 

3.   Favard J. Sur l’interpolation. J. Math. Pures Appl., 1940, vol. 19, no. 9, pp. 281–306.

4.   de Boor С. How small can one make the derivatives of an interpolating function? J. Approx. Theory, 1975, vol. 13, no. 2, pp. 105–116. doi: 10.1016/0021-9045(75)90043-X 

5.   de Boor С. On “best” interpolation J. Approx. Theory, 1976, vol. 16, no. 1, pp. 28–42. doi: 10.1016/0021-9045(76)90093-9 

6.   Pevnyi A.B. Natural splines of two and three variables. Metody Vychisl., 1985, vol. 14, pp. 160–170 (in Russian).

7.   Smolyak S.A. Optimal recovery of functions and related geometric characteristics of sets. Proc. 3th Winter Workshop on Mathematical Programming, Moscow: TsEMI AN SSSR Publ., 1971, pp. 509–557 (in Russian).

8.   Duchon J. Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. RAIRO Anal. Numer., 1976, vol. 10, no. R3, pp. 5–12. doi: 10.1051/m2an/197610R300051 

9.   Arcangeli R., de Silanes M.C., Torrens Ju. Multidimensional minimizing splines. Theory and Applications. N.Y. etc.: Kluwer Acad. Publ., 2004, 261 p. ISBN: 1402077866 .

10.   Buhmann M.D. Radial basis functions. Acta Numer., 2000, vol. 9, pp. 1–38. doi: 10.1017/S0962492900000015 

11.   Buhmann M.D. Radial basis functions: theory and implementations. Cambridge Monogr. Appl. Comput. Math., no. 12. Cambridge: Cambridge Univ. Press, 2003, 259 p. ISBN: 0521633389 .

12.   Mehri B., Jokar S. Lebesgue function for multivariate interpolation by radial basis functions. Appl. Math. Comput., 2007, vol. 187, no. 1, pp. 306–314. doi: 10.1016/j.amc.2006.08.127 

13.   Fikhtengol’ts G.M. Kurs differentsial’nogo i integral’nogo ischisleniya [Course of differential and integral calculus]. Vol. 1. Мoscow: Nauka Publ., 1970, 608 p.

14.   Holmes R.B. Geometric functional analysis and its applications. NY: Springer, 1975, 246 p. doi: 10.1007/978-1-4684-9369-6 

Cite this article as: S.I. Novikov. On an interpolation problem with the smallest L2-norm of the Laplace operator. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 143–153; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S193–S203.