Geodesic
The Helly problem and best approximation of summable functions
Sbornik. Mathematics, Tome 13 (1971) no. 2, pp. 187-207 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Approximative properties are studied for the subspaces $\{L^n\}$ of finite codimensionality $n$ in the space of summable functions $L_1=L_1(T,\Sigma,\mu)$. Criteria are established for a subspace in which for every $x\in L_1$ there exists an element of best approximation (or a unique such element). A dual interpretation of these results in terms of the existence and uniqueness of minimal solutions of the finite problem of moments (“Helly problem”) is given. The question of construction of generalized elements of best approximation in a certain extension of the space $L_1$ is considered. Bibliography: 18 titles. 
@article{SM_1971_13_2_a1,
     author = {A. L. Garkavi},
     title = {The {Helly} problem and best approximation of summable functions},
     journal = {Sbornik. Mathematics},
     pages = {187--207},
     year = {1971},
     volume = {13},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_13_2_a1/}
}
TY  - JOUR
AU  - A. L. Garkavi
TI  - The Helly problem and best approximation of summable functions
JO  - Sbornik. Mathematics
PY  - 1971
SP  - 187
EP  - 207
VL  - 13
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1971_13_2_a1/
LA  - en
ID  - SM_1971_13_2_a1
ER  - 
%0 Journal Article
%A A. L. Garkavi
%T The Helly problem and best approximation of summable functions
%J Sbornik. Mathematics
%D 1971
%P 187-207
%V 13
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1971_13_2_a1/
%G en
%F SM_1971_13_2_a1
A. L. Garkavi. The Helly problem and best approximation of summable functions. Sbornik. Mathematics, Tome 13 (1971) no. 2, pp. 187-207. http://geodesic.mathdoc.fr/item/SM_1971_13_2_a1/

[1] N. I. Akhiezer, M. G. Krein, “$L$-problema momentov”, V kn: N. I. Akhiezer, M. G. Krein, Nekotorye voprosy teorii momentov, GONTI, Kharkov, 1938

[2] S. N. Bernshtein, Ekstremalnye svoistva polinomov, ONTI, Moskva–Leningrad, 1937

[3] N. Burbaki, Topologicheskie vektornye prostranstva, IL, Moskva, 1959

[4] A. L. Garkavi, “Teoremy dvoistvennosti dlya priblizhenii posredstvom elementov vypuklykh mnozhestv”, Uspekhi matem. nauk, XVI:4(100) (1961), 141–145 | MR

[5] A. L. Garkavi, “O nailuchshem priblizhenii elementami podprostranstv odnogo klassa”, Matem. sb., 62(104) (1963), 104–120 | MR | Zbl

[6] A. L. Garkavi, “O chebyshevskikh podprostranstvakh konechnogo defekta v prostranstve summiruemykh funktsii”, Uchenye zapiski Orekhovo-Zuevskogo ped. in-ta, 22:3 (1964), 5–11 | MR

[7] A. L. Garkavi, “O edinstvennosti resheniya $L$-problemy momentov”, Izv. AN SSSR, seriya matem., 28 (1964), 553–570 | MR | Zbl

[8] A. L. Garkavi, “Zadacha Khelli i nailuchshee priblizhenie v prostranstve nepreryvnykh funktsii”, Izv. AN SSSR, seriya matem., 31 (1967), 641–657 | MR

[9] A. L. Garkavi, “O kompaktakh, dopuskayuschikh chebyshevskie sistemy mer”, Matem. sb., 74(116) (1967), 209–217 | MR | Zbl

[10] N. Danford, Dzh. Shvarts, Lineinye operatory, IL, Moskva, 1962

[11] J. Singer, “Asupra $L$-problemei teoriei momentelor in spajii Banach”, Bui. Stunf. Acad. R. P. R., Sec. Mat. fiz., 9:1 (1957), 19–28 | Zbl

[12] J. Singer, Cea mai buna approximare in spatii vectoriale, Acad. R. S. R., Bucuresti, 1967 | Zbl

[13] M. G. Krein, “$L$-problema v abstraktnom lineinom normirovannom prostranstve”, V kn.: N. I. Akhiezer, M. G. Krein, O nekotorykh voprosakh teorii momentov, GONTI, Kharkov, 1938

[14] L. V. Taikov, “Odin krug ekstremalnykh zadach dlya trigonometricheskikh polinomov”, Uspekhi matem. nauk, XX:3(123) (1965), 205–211 | MR

[15] R. Felps, Lektsii o teoremakh Shoke, Mir, Moskva, 1968

[16] R. R. Phelps, “Uniqueness of Hahn–Banach extensions and unique best approximation”, Trans. Amer. Math. Soc., 95:2 (1960), 238–255 | DOI | MR | Zbl

[17] E. HeLLy, “Ober Systeme linearer Gleichungen mit unendlich vielen Unbekannten”, Monatsh. Math. Phys., 31 (1921), 60–91 | DOI | MR | Zbl

[18] P. D. Morris, “Metric projections onto subspaces of finite codimension”, Duke Math. J., 35:4 (1968), 799–808 | DOI | MR | Zbl