We consider spaces of entire functions with systems of weighted estimates. The case of twoterm weight sequences consisting of radial and nonradial components is studied. 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.
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References
V. V. Napalkov, Convolution Equations in Multidimensional Spaces [in Russian], Moscow (1982).
R. W. Braun, R. Meise, and B. A. Taylor, “Ultradifferentiable functions and Fourier analysis,” Res. Math., 17, 206–237 (1990).
R. Meise, B. A. Taylor, and D. Vogt, “Equivalence of slowly decreasing conditions and local Fourier expansions,” Indiana Univ. Math. J., 36, 729–756 (1987).
T. Meyer, “Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type,” Studia Math., 125, 101–129 (1997).
D. A. Abanina, “On Borel’s theorem for spaces of ultradifferentiable functions of mean type,” Res. Math., 44, 195–213 (2003).
A. V. Abanin and Pham Trong Tien, “Almost subadditive weight functions form Braun–Meise–Taylor theory of ultradistributions,” J. Math. Anal. Appl., 363, 296–301 (2010).
S. Momm, “Closed principal ideals in nonradial Hörmander algebras,” Arch. Math. (Basel), 58, 47–55 (1992).
A. V. Abanin and Le Hai Khoi, “Pre-dual of the function algebra A −∞(D) and representation of functions in Dirichlet series,” Complex Anal. Oper. Theory, 5, 1073–1092 (2011).
L. Carleson, “Interpolation by bounded analytic functions and the corona problem,” Ann. Math., 72, 547–559 (1962).
J. J. Kelleher and B. A. Taylor “An application of the corona theorem to some rings of entire functions,” Bull. Amer. Math. Soc., 73, 246–249 (1967).
L. Hörmander, An Introduction to Complex Analysis in Several Variables [Russian translation], Moscow (1968).
L. Hörmander, “Generators for some rings of analytic functions,” Bull. Amer. Math. Soc., 73, 943–949 (1967).
T. T. Kuzbekov, “On ideals in some algebras of analytic functions,” Izv. VUZ, Matem., 10, 77–80 (1993).
T. T. Kuzbekov, “On generators in some rings of analytic functions,” Mat. Zametki, 55, 68–75 (1994).
A. V. Abanin, “Modification of L. Hörmander’s method in the problem of generators and its applications,” Izv. VUZ, Matem., 8, 3–12 (1995).
O. V. Epifanov, “Solvability of the inhomogeneous Cauchy–Riemann equation in classes of functions that are bounded with weight and a system of weights,” Mat. Zametki, 51, 83–92 (1992).
A. V. Abanin and Pham Trong Tien, “Continuation of holomorphic functions and some of its applications,” Studia Math., 200, 279–295 (2010).
D. A. Polyakova, “Solvability of inhomogeneous Cauchy–Riemann equation in spaces of functions with a system of uniform weight estimates,” Izv. VUZ, Matem., 10, 77–82 (2015).
A. V. Abanin and I. A. Filip’ev, “Analytic realization of spaces dual to spaces of infinitely differentiable functions,” Sib. Mat. Zh., 47, 485–500(2006).
D. A. Abanina, “Solvability of convolution equation in Beurling spaces of ultradifferentiable functions of normal type on an interval,” Sib. Mat. Zh., 53, 477–494 (2012).
R. Edwards, Functional Analysis [Russian translation], Moscow (1969).
A. P. Robertson and W. Robertson, Topological Vector Spaces [Russian translation], Moscow (1967).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 447, 2016, pp. 90–112.
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Polyakova, D.A. Generators for Spaces of Entire Functions with a System of Weighted Estimates. J Math Sci 229, 545–560 (2018). https://doi.org/10.1007/s10958-018-3697-0
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DOI: https://doi.org/10.1007/s10958-018-3697-0