Geodesic
Extremal functional $L_p$-interpolation on an arbitrary mesh on the real axis
Sbornik. Mathematics, Tome 213 (2022) no. 4, pp. 556-577 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Golomb-de Boor problem of extremal interpolation of infinite real sequences with smallest $L_p$-norm of the $n$th derivative of the interpolant, $1\le p\le \infty$, on an arbitrary mesh on the real axis is studied under constraints on the norms of the corresponding divided differences. For this smallest norm, lower estimates are obtained for any $n\in \mathbb N$ in terms of $B$-splines. For the second derivative, this quantity is estimated from below and above by constants depending on the parameter $p$. Bibliography: 13 titles.
Keywords: extremal interpolation, derivative, divided difference, spline, difference equation. 
@article{SM_2022_213_4_a5,
     author = {Yu. N. Subbotin and V. T. Shevaldin},
     title = {Extremal functional $L_p$-interpolation on an arbitrary mesh on the real axis},
     journal = {Sbornik. Mathematics},
     pages = {556--577},
     year = {2022},
     volume = {213},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2022_213_4_a5/}
}
TY  - JOUR
AU  - Yu. N. Subbotin
AU  - V. T. Shevaldin
TI  - Extremal functional $L_p$-interpolation on an arbitrary mesh on the real axis
JO  - Sbornik. Mathematics
PY  - 2022
SP  - 556
EP  - 577
VL  - 213
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_2022_213_4_a5/
LA  - en
ID  - SM_2022_213_4_a5
ER  - 
%0 Journal Article
%A Yu. N. Subbotin
%A V. T. Shevaldin
%T Extremal functional $L_p$-interpolation on an arbitrary mesh on the real axis
%J Sbornik. Mathematics
%D 2022
%P 556-577
%V 213
%N 4
%U http://geodesic.mathdoc.fr/item/SM_2022_213_4_a5/
%G en
%F SM_2022_213_4_a5
Yu. N. Subbotin; V. T. Shevaldin. Extremal functional $L_p$-interpolation on an arbitrary mesh on the real axis. Sbornik. Mathematics, Tome 213 (2022) no. 4, pp. 556-577. http://geodesic.mathdoc.fr/item/SM_2022_213_4_a5/

[1] M. Golomb, “$H^{m,p}$-extensions by $H^{m,p}$-splines”, J. Approximation Theory, 5:3 (1972), 238–275 | DOI | MR | Zbl

[2] C. de Boor, “How small can one make the derivatives of an interpolating function?”, J. Approximation Theory, 13:2 (1975), 105–116 | DOI | MR | Zbl

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

[4] 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

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

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

[7] 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

[8] C. de Boor, “A smooth and local interpolant with “small” $k$-th derivative”, Numerical solutions of boundary value problems for ordinary differential equations (Univ. Maryland, Baltimore, MD, 1974), Academic Press, New York, 1975, 177–197 | MR | Zbl

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

[10] 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

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

[12] S. I. Novikov, V. T. Shevaldin, “Ekstremalnaya interpolyatsiya na poluosi s naimenshim znacheniem normy tretei proizvodnoi”, Tr. IMM UrO RAN, 26, no. 4, 2020, 210–223 | DOI | MR

[13] Yu. S. Zavyalov, B. I. Kvasov, V. L. Miroshnichenko, Metody splain-funktsii, Nauka, M., 1980, 352 pp. | MR | Zbl