Geodesic
Approximation of Müntz-Szász type in weighted spaces
Sbornik. Mathematics, Tome 204 (2013) no. 7, pp. 1028-1055 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper looks at whether a system of exponentials $\exp(-\lambda_nt)$, $\operatorname{Re}\lambda_n>0$, is complete in various function spaces on the half-line $\mathbb R_+$. Wide classes of Banach spaces $E$ and $F$ of functions on $\mathbb R_+$ are described such that this system is complete in $E$ and $F$ simultaneously. A test is established to determine when this system is complete in the weighted spaces $C_0$ and $L^p$ with weight $(1+t)^\alpha$ on $\mathbb R_+$, for $\alpha<0$ and $\alpha<-1$, respectively. Bibliography: 18 titles.
Keywords: Müntz and Szász theorems, complete system of exponentials, spaces with combined norm, weighted spaces
Mots-clés : Laplace transform. 
@article{SM_2013_204_7_a4,
     author = {A. M. Sedletskii},
     title = {Approximation of {M\"untz-Sz\'asz} type in weighted spaces},
     journal = {Sbornik. Mathematics},
     pages = {1028--1055},
     year = {2013},
     volume = {204},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_7_a4/}
}
TY  - JOUR
AU  - A. M. Sedletskii
TI  - Approximation of Müntz-Szász type in weighted spaces
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 1028
EP  - 1055
VL  - 204
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_7_a4/
LA  - en
ID  - SM_2013_204_7_a4
ER  - 
%0 Journal Article
%A A. M. Sedletskii
%T Approximation of Müntz-Szász type in weighted spaces
%J Sbornik. Mathematics
%D 2013
%P 1028-1055
%V 204
%N 7
%U http://geodesic.mathdoc.fr/item/SM_2013_204_7_a4/
%G en
%F SM_2013_204_7_a4
A. M. Sedletskii. Approximation of Müntz-Szász type in weighted spaces. Sbornik. Mathematics, Tome 204 (2013) no. 7, pp. 1028-1055. http://geodesic.mathdoc.fr/item/SM_2013_204_7_a4/

[1] Ch. H. Müntz, “Über den Approximationssatz Weierstrass”, Schwartz Festschrift, Berlin, 1914, 303–312 | Zbl

[2] O. Szász, “Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen”, Math. Ann., 77:4 (1916), 482–496 | DOI | MR | Zbl

[3] N. I. Achieser, Theory of approximation, Ungar Publ., New York, 1956 | MR | MR | Zbl | Zbl

[4] L. Schwartz, Étude des sommes d'exponentielles, Hermann, Paris, 1959 | MR | Zbl

[5] M. M. Grum, “On the theorems of Müntz and Szász”, J. London Math. Soc., 31 (1956), 433–437 | DOI | MR | Zbl

[6] M. M. Grum, “Corrigendum and Addendum: On the theorems of Müntz and Szász”, J. London Math. Soc., 32 (1957), 512 | DOI | Zbl

[7] A. R. Siegel, “On the Müntz–Szász theorem for C$[0,1]$”, Proc. Amer. Math. Soc., 36:1 (1972), 161–166 | MR | Zbl

[8] N. Levinson, “On the Szász–Müntz theorem”, J. Math. Anal. Appl., 48:1 (1974), 264–269 | DOI | MR | Zbl

[9] A. M. Sedletskii, “K probleme Myuntsa–Sasa dlya prostranstva $C[0,1]$”, Tr. MEI, 1975, no. 260, 89–98

[10] V. I. Ladygin, “The Müntz–Szasz problem”, Math. Notes, 23:1 (1978), 51–58 | DOI | MR | Zbl | Zbl

[11] A. M. Sedletskii, Klassy analiticheskikh preobrazovanii Fure i eksponentsialnye approksimatsii, Fizmatlit, M., 2005

[12] A. M. Sedletskii, “Müntz–Szász type approximation in direct products of spaces”, Izv. Math., 70:5 (2006), 1031–1050 | DOI | DOI | MR | Zbl

[13] G. F. Votruba, L. F. Boron (eds.), Functional analysis, Wolters-Noordhoff Publ., Groningen, 1972 | MR | MR | Zbl

[14] B. N. Khabibullin, “Polnota sistem eksponent v prostranstvakh funktsii na luche”, Ufimsk. matem. zhurn., 1:1 (2009), 77–83 | Zbl

[15] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970 | MR | MR | Zbl | Zbl

[16] H. S. Shapiro, A. L. Shields, “On the zeros of functions with finite Dirichlet integral and some related function spaces”, Math. Z., 80:1 (1962), 217–229 | DOI | MR | Zbl

[17] I. I. Privalov, Granichnye svoistva analiticheskikh funktsii, GITTL, M.–L., 1950 | MR | Zbl

[18] G. M. Goluzin, Geometric theory of functions of a complex variable, Transl. Math. Monogr., 26, Amer. Math. Soc., Providence, RI, 1969 | MR | MR | Zbl | Zbl