Geodesic
Characterization of Hadamard vector classes in terms of least deviations of their elements from vectors of finite degree
Sbornik. Mathematics, Tome 192 (2001) no. 12, pp. 1829-1876 
G. V. Radzievskii. Characterization of Hadamard vector classes in terms of least deviations of their elements from vectors of finite degree. Sbornik. Mathematics, Tome 192 (2001) no. 12, pp. 1829-1876. http://geodesic.mathdoc.fr/item/SM_2001_192_12_a3/
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     author = {G. V. Radzievskii},
     title = {Characterization of {Hadamard} vector classes in terms of least deviations of their elements from vectors of finite degree},
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     url = {http://geodesic.mathdoc.fr/item/SM_2001_192_12_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $A$ be a linear operator with domain $\mathfrak D(A)$ in a complex Banach space $X$. An element $g\in\mathfrak D_\infty(A):=\bigcap_{j=0}^\infty\mathfrak D(A^j)$ is called a vector of degree at most $\xi$ $(>0)$ relative to $A$ if $\|A^jg\|\leqslant c(g)\xi^j$, $j=0,1,\dots$ . The set of vectors of degree at most $\xi$ is denoted by $\mathfrak G_\xi(A)$ and the least deviation of an element $f$ of $X$ from the set $\mathfrak G_\xi(A)$ is denoted by $E_\xi(f,A)$. For a fixed sequence of positive numbers $\{\psi_j\}_{j=1}^\infty$ consider a function $\gamma(\xi):=\min_{j=1,2,\dots}(\xi\psi_j)^{1/j}$. Conditions for the sequence $\{\psi_j\}_{j=1}^\infty$ and the operator $A$ are found that ensure the equality $$ \limsup_{j\to\infty}\biggl(\frac{\|A^jf\|}{\psi_j}\biggr)^{1/j}=\limsup_{\xi\to\infty}\frac\xi{\gamma(E_\xi(f,A)^{-1})}\,. $$ for $f\in\mathfrak D_\infty(A)$. If the quantity on the left-hand side of this formula is finite, then $f$ belongs to the Hadamard class determined by the operator $A$ and the sequence $\{\psi_j\}_{j=1}^\infty$. One consequence of the above formula is an expression in terms of $E_\xi(f,A)$ for the radius of holomorphy of the vector-valued function $F(zA)f$, where $f\in\mathfrak D_\infty(A)$, and $F(z):=\sum_{j=1}^\infty z^j/\psi_j$ is an entire function.

[1] Hadamard J., “Sur la généralisation de la notion de fonction analytique”, Bull. Soc. Math., Comptes Rendus des Séances, 40 (1912), 28

[2] Vallée Poussin Ch.-J., Leçons sur l'approximation des fonctions continues, Paris, 1918

[3] Bernshtein S. N., Sobranie sochinenii, T. II, Izd-vo AN SSSR, M., 1954

[4] Mandelbroit S., Primykayuschie ryady. Regulyarizatsiya posledovatelnostei. Primeneniya, IL, M., 1955

[5] Mandelbroit S., Kvazianaliticheskie klassy funktsii, Gostekhizdat, M.-L., 1937

[6] Polia G., Sege G., Zadachi i teoremy iz analiza, T. 2, Gostekhizdat, M., 1956

[7] Radzievskii G. V., “Ob odnom obobschenii formuly Bernshteina dlya opredeleniya shiriny polosy golomorfnosti funktsii”, Funkts. analiz i ego prilozh., 33:2 (1999), 43–57 | MR | Zbl

[8] Bernshtein S. N., Sobranie sochinenii, T. I, Izd-vo AN SSSR, M., 1952

[9] Stechkin S. B., “O poryadke nailuchshikh priblizhenii nepreryvnykh funktsii”, Izv. AN SSSR. Ser. matem., 15:3 (1951), 219–242 | MR | Zbl

[10] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960

[11] Radzievskii G. V., “Pryamye i obratnye teoremy v zadachakh o priblizhenii po vektoram konechnoi stepeni”, Matem. sb., 189:4 (1998), 83–124 | MR | Zbl

[12] Gorbachuk M. L., Gorbachuk V. I., “Pro nablizhennya gladkikh vektoriv zamknenogo operatora tsilimi vektorami eksponentsialnogo tipu”, Ukr. matem. zhurn., 47:5 (1995), 616–628 | MR | Zbl

[13] Gorbachuk V. I., Gorbachuk M. L., “Operatornyi podkhod k zadacham approksimatsii”, Algebra i analiz, 9:6 (1997), 90–108 | MR | Zbl

[14] Goldberg A. A., Levin B. Ya., Ostrovskii I. V., “Tselye i meromorfnye funktsii”, Itogi nauki i tekhniki. Sovrem. probl. matem. Fundam. napr., 85, VINITI, M., 1991, 5–185 | MR

[15] Evgrafov M. A., Asimptoticheskie otsenki i tselye funktsii, Fizmatgiz, M., 1962 | MR

[16] Fridman G. A., “Medlenno vozrastayuschie funktsii i ikh prilozhenie”, Sib. matem. zhurn., 7:5 (1966), 1139–1160 | MR | Zbl

[17] Radzievskii G. V., “O nailuchshikh priblizheniyakh i o skorosti skhodimosti razlozhenii po kornevym vektoram operatora”, Ukr. matem. zhurn., 49:6 (1997), 754–773 | MR

[18] Kuptsov N. P., “Pryamye i obratnye teoremy teorii priblizhenii i polugruppy operatorov”, UMN, 23:4 (1968), 117–178 | MR | Zbl

[19] Butzer P. L., Scherer K., “Jackson and Bernstein-type inequalities for families of commutative operators in Banach spaces”, J. Approx. Theory, 5:3 (1972), 308–342 | DOI | MR | Zbl

[20] Gorin E. A., “Neravenstva Bernshteina s tochki zreniya teorii operatorov”, Vestn. Khark. un-ta. Prikl. matem., mekh., 205:45 (1980), 77–105 | MR | Zbl

[21] Gomilko A. M., Radzievskii G. V., “Bazisnye svoistva sobstvennykh funktsii regulyarnoi kraevoi zadachi dlya vektornogo funktsionalno-differentsialnogo uravneniya”, Differents. uravneniya, 27:3 (1991), 384–396 | MR | Zbl

[22] Radzievskii G., “The rate of convergence of decompositions of ordinary functional-differential operators by eigenfunctions”, Some problems of the modern theory of differential equations, Kiev, 1994, 14–27 ; Preprint 94.29, Nat. Acad. Sci. Ukr. Inst. Math. | MR | Zbl

[23] Radzievskii G. V., “Kraevye zadachi i svyazannye s nimi moduli nepreryvnosti”, Funkts. analiz i ego prilozh., 29:3 (1995), 87–90 | MR | Zbl

[24] Radzievskii G. V., “Neravenstva Dzheksona i Bernshteina dlya sistemy kornevykh funktsii operatora differentsirovaniya s nelokalnym kraevym usloviem”, Dokl. RAN, 363:1 (1998), 20–23 | MR | Zbl

[25] Seneta E., Pravilno menyayuschiesya funktsii, Nauka, M., 1985 | MR | Zbl

[26] Galamlos J., Seneta E., “Regularly varying sequences”, Proc. Amer. Math. Soc., 41:1 (1973), 110–116 | DOI | MR

[27] Bojanić R., Seneta E., “A unified theory of regularly varying sequences”, Math. Z., 134 (1973), 91–106 | DOI | MR | Zbl

[28] Jackson D., Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung, Inauguraldissertation und Preischrift, Univ. Göttingen, 1911 | Zbl

[29] Danford N., Shvarts Dzh. T., Lineinye operatory. Obschaya teoriya, IL, M., 1962

[30] Khille E., Fillips R., Funktsionalnyi analiz i polugruppy, IL, M., 1962

[31] Nelson E., “Analytic vector”, Ann. of Math. (2), 70:3 (1959), 572–615 | DOI | MR | Zbl

[32] Goodman R., “Analytic and entire vectors for representations of Lie groups”, Trans. Amer. Math. Soc., 143 (1969), 55–76 | DOI | MR | Zbl

[33] Gorbachuk V. I., Gorbachuk M. L., Granichnye zadachi dlya differentsialno-operatornykh uravnenii, Naukova dumka, Kiev, 1984 | MR | Zbl

[34] Goldberg A. A., Ostrovskii I. V., Raspredelenie znachenii meromorfnykh funktsii, Nauka, M., 1970 | MR

[35] Markushevich A. I., Teoriya analiticheskikh funktsii, T. 2, Nauka, M., 1968 | Zbl