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To the question on \(\gamma\)-sufficient sets. (English. Russian original) Zbl 0970.32002

Sib. Math. J. 41, No. 4, 778-784 (2000); translation from Sib. Mat. Zh. 41, No. 4, 935-943 (2000).
Weakly sufficient sets, introduced by D. Schneider, play an important role in the questions of representing analytic functions by function series. Their natural generalization, the so-called \(\gamma\)-sufficient sets, were proposed by Yu. Korobeinik who, in particular, applied them to constructing discrete maximum sets in some classes of entire functions (a Levinson-type theorem). The question of interrelation between weakly sufficient sets and (Iyer) effective sets was discussed by Yu. Korobeinik [Russ. Math. Surv. 36, No. 1, 75-137 (1981); translation from Usp. Mat. Nauk 36, No. 1 (21), 73-126 (1981; Zbl 0483.30003)] and completely in A. B. Abanin [Mat. Zametki 40, No. 4, 442-454 (1986; Zbl 0617.46032)].
In the article under review, the author uses the method suggested by A. B. Abanin (involving more general results) to solve a similar problem for \(\gamma\)-sufficient sets.

MSC:

32A15 Entire functions of several complex variables
32K05 Banach analytic manifolds and spaces
32A17 Special families of functions of several complex variables
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References:

[1] Schneider D. M., ”Sufficient sets for some spaces of entire functions,” Trans. Amer. Math. Soc.,197, 161–180 (1974). · Zbl 0264.46018 · doi:10.1090/S0002-9947-1974-0357835-2
[2] Korobeînik Yu. F., ”Representing systems,” Uspekhi Mat. Nauk,36, No. 1, 73–126 (1981).
[3] Korobeînik Yu. F., ”Inductive and projective topologies. Sufficient sets and representing systems,” Izv. Akad. Nauk SSSR Ser. Mat.,50, No. 3, 539–565 (1986).
[4] Abanin A. V., ”On some criteria for weak sufficiency,” Mat. Zametki,40, No. 4, 442–454 (1986). · Zbl 0617.46032
[5] Korobeînik Yu. F., ”Maximal and {\(\gamma\)}-sufficient sets. Applications to entire functions. I,” in: Function Theory, Functional Analysis, and Some of Their Applications [in Russian], 1990, No. 54, pp. 42–59. · Zbl 0741.30023
[6] Sherstyukov V. B., ”On a certain class of complete systems,” Izv. Vyssh. Uchebn. Zaved. Estestv. Nauki,2, 38–40 (1997). · Zbl 0918.46007
[7] Edwards R. D., Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969). · Zbl 0189.12103
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