On generators of spaces of entire functions with a system of weighted estimates

D. A. Polyakova

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 90-112

Résumé

We consider spaces of entire functions with systems of weight estimates. The case of binomial weight sequences consisting of radial and nonradial components is investigated. Under some assumptions on the weight sequence we obtain a complete description of generators in these spaces. We apply this result to the problem of normal solvability of systems of convolution equations in the Roumieu spaces of ultradifferentiable functions and, as a particular case, in Gevrey classes.

Bibliographie

[1] V. V. Napalkov, Uravneniya svertki v mnogomernykh prostranstvakh, Nauka, M., 1982 | MR

[2] R. W. Braun, R. Meise, B. A. Taylor, “Ultradifferentiable functions and Fourier analysis”, Results Math., 17 (1990), 206–237 | DOI | MR | Zbl

[3] R. Meise, B. A. Taylor, D. Vogt, “Equivalence of slowly decreasing conditions and local Fourier expansions”, Indiana Univ. Math. J., 36 (1987), 729–756 | DOI | MR | Zbl

[4] T. Meyer, “Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type”, Studia Math., 125 (1997), 101–129 | MR | Zbl

[5] D. A. Abanina, “On Borel's theorem for spaces of ultradifferentiable functions of mean type”, Res. Math., 44 (2003), 195–213 | DOI | MR | Zbl

[6] A. V. Abanin, Pham Trong Tien, “Almost subadditive weight functions form Braun–Meise–Taylor theory of ultradistributions”, J. Math. Anal. Appl., 363 (2010), 296–301 | DOI | MR | Zbl

[7] S. Momm, “Closed principal ideals in nonradial Hörmander algebras”, Arch. Math. (Basel), 58 (1992), 47–55 | DOI | MR | Zbl

[8] A. V. Abanin, Le Hai Khoi, “Pre-dual of the function algebra $A^{-\infty}(D)$ and representation of functions in Dirichlet series”, Complex Anal. Oper. Theory, 5:4 (2011), 1073–1092 | DOI | MR | Zbl

[9] L. Carleson, “Interpolation by bounded analytic functions and the corona problem”, Ann. of Math., 72:2 (1962), 547–559 | DOI | MR

[10] J. J. Kelleher, B. A. Taylor, “An application of the corona theorem to some rings of entire functions”, Bull. Amer. Math. Soc., 73 (1967), 246–249 | DOI | MR | Zbl

[11] L. Khërmander, Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR

[12] L. Hörmander, “Generators for some rings of analytic functions”, Bull. Amer. Math. Soc., 73 (1967), 943–949 | DOI | MR | Zbl

[13] T. T. Kuzbekov, “Ob idealakh v nekotorykh koltsakh analiticheskikh funktsii”, Izv. vuzov. Matem., 1993, no. 10, 77–80 | MR | Zbl

[14] T. T. Kuzbekov, “K voprosu o porozhdayuschikh v nekotorykh koltsakh analiticheskikh funktsii”, Matem. zametki, 55:3 (1994), 68–75 | MR | Zbl

[15] A. V. Abanin, “Modifikatsiya metoda L. Khërmandera v zadache o porozhdayuschikh i ee prilozheniya”, Izv. vuzov. Matem., 1995, no. 8, 3–12 | MR | Zbl

[16] O. V. Epifanov, “O razreshimosti neodnorodnogo uravneniya Koshi–Rimana v klassakh funktsii, ogranichennykh s vesom i sistemoi vesov”, Mat. zametki, 51:1 (1992), 83–92 | MR | Zbl

[17] A. V. Abanin, Pham Trong Tien, “Continuation of holomorphic functions and some of its applications”, Studia Math., 200 (2010), 279–295 | DOI | MR | Zbl

[18] D. A. Polyakova, “O razreshimosti neodnorodnogo uravneniya Koshi–Rimana v prostranstvakh funktsii s sistemoi ravnomernykh vesovykh otsenok”, Izv. vuzov. Matem., 2015, no. 10, 77–82 | Zbl

[19] A. V. Abanin, I. A. Filipev, “Analiticheskaya realizatsiya prostranstv, sopryazhennykh k prostranstvam beskonechno differentsiruemykh funktsii”, Sib. mat. zhurn., 47:3 (2006), 485–500 | MR | Zbl

[20] D. A. Abanina, “Razreshimost uravnenii svertki v prostranstvakh ultradifferentsiruemykh funktsii Berlinga normalnogo tipa na intervale”, Sib. mat. zhurn., 53:3 (2012), 477–494 | MR | Zbl

[21] R. Edvards, Funktsionalnyi analiz, Mir, M., 1969

[22] A. P. Robertson, V. Dzh. Robertson, Topologicheskie vektornye prostranstva, Mir, M., 1967 | MR