Geodesic
Extremal interpolation with the least value of~the norm of~the second derivative in~$L_p(\mathbb R)$
Izvestiya. Mathematics , Tome 86 (2022) no. 1, pp. 203-219.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we formulate a general problem of extreme functional interpolation of real-valued functions of one variable (for finite differences, this is the Yanenko–Stechkin–Subbotin problem) in terms of divided differences. The least value of the $n$-th derivative in $L_p(\mathbb R)$, $1\le p\le \infty$, needs to be calculated over the class of functions interpolating any given infinite sequence of real numbers on an arbitrary grid of nodes, infinite in both directions, on the number axis $\mathbb R$ for the class of interpolated sequences for which the sequence of $n$-th order divided differences belongs to $l_p(\mathbb Z)$. In the present paper this problem is solved in the case when $n=2$. The indicated value is estimated from above and below using the greatest and the least step of the grid of nodes.
Keywords: divided difference, spline, difference equation.
Mots-clés : interpolation 
@article{IM2_2022_86_1_a6,
     author = {V. T. Shevaldin},
     title = {Extremal interpolation with the least value of~the norm of~the second derivative in~$L_p(\mathbb R)$},
     journal = {Izvestiya. Mathematics },
     pages = {203--219},
     publisher = {mathdoc},
     volume = {86},
     number = {1},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2022_86_1_a6/}
}
TY  - JOUR
AU  - V. T. Shevaldin
TI  - Extremal interpolation with the least value of~the norm of~the second derivative in~$L_p(\mathbb R)$
JO  - Izvestiya. Mathematics 
PY  - 2022
SP  - 203
EP  - 219
VL  - 86
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2022_86_1_a6/
LA  - en
ID  - IM2_2022_86_1_a6
ER  - 
%0 Journal Article
%A V. T. Shevaldin
%T Extremal interpolation with the least value of~the norm of~the second derivative in~$L_p(\mathbb R)$
%J Izvestiya. Mathematics 
%D 2022
%P 203-219
%V 86
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2022_86_1_a6/
%G en
%F IM2_2022_86_1_a6
V. T. Shevaldin. Extremal interpolation with the least value of~the norm of~the second derivative in~$L_p(\mathbb R)$. Izvestiya. Mathematics , Tome 86 (2022) no. 1, pp. 203-219. http://geodesic.mathdoc.fr/item/IM2_2022_86_1_a6/

[1] A. O. Gelfond, Calculus of finite differences, Int. Monogr. Adv. Math. Phys., Hindustan Publishing Corp., Delhi, 1971, vi+451 pp. | MR | MR | Zbl | Zbl

[2] S. B. Stechkin, Yu. N. Subbotin, Splainy v vychislitelnoi matematike, Nauka, M., 1976, 248 pp. | MR | Zbl

[3] J. Favard, “Sur l'interpolation”, J. Math. Pures Appl. (9), 19:9 (1940), 281–306 | MR | Zbl

[4] V. S. Ryaben'kii, “Necessary and sufficient conditions for good definition of boundary value problems for systems of ordinary difference equations”, U.S.S.R. Comput. Math. Math. Phys., 4:2 (1964), 43–61 | DOI | MR | Zbl

[5] V. S. Rjabenki, A. F. Filippow, Über die Stabilität von Differenzengleichungen, Mathematik für Naturwissenschaft und Technik, 3, VEB Deutscher Verlag der Wissenschaften, Berlin, 1960, viii+136 pp. | MR | MR | Zbl | Zbl

[6] S. L. Sobolev, Lektsii po teorii kubaturnykh formul, Ch. 2, Izd-vo Novosibirskogo un-ta, Novosibirsk, 1965, 293 pp. | MR

[7] Yu. N. Subbotin, “O svyazi mezhdu konechnymi raznostyami i sootvetstvuyuschimi proizvodnymi”, Ekstremalnye svoistva polinomov, Sbornik rabot, Tr. MIAN SSSR, 78, Nauka, M., 1965, 24–42 | MR | Zbl

[8] Yu. N. Subbotin, “Functional interpolation in the mean with smallest $n$ derivative”, Proc. Steklov Inst. Math., 88 (1967), 31–63 | MR | Zbl

[9] Yu. N. Subbotin, “Extremal problems of functional interpolation, and mean interpolation splines”, Proc. Steklov Inst. Math., 138 (1977), 127–185 | MR | Zbl

[10] Yu. N. Subbotin, S. I. Novikov, V. T. Shevaldin, “Ekstremalnaya funktsionalnaya interpolyatsiya i splainy”, Tr. IMM UrO RAN, 24, no. 3, 2018, 200–225 | DOI | MR

[11] Th. Kunkle, “Favard's interpolation problem in one or more variables”, Constr. Approx., 18:4 (2002), 467–478 | DOI | MR | Zbl

[12] S. I. Novikov, V. T. Shevaldin, “O svyazi mezhdu vtoroi razdelennoi raznostyu i vtoroi proizvodnoi”, Tr. IMM UrO RAN, 26, no. 2, 2020, 216–224 | DOI | MR