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References
Bagby, T. and Gauthier, P. M.Harmonic approximation on closed subsets of Riemannian manifolds, Complex potential theory (Montreal, PQ, 1993), 75–87, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Kluwer Acad. Publ., Dordrecht, 1994.
Bagemihl, F. and Seidel, W.Some boundary properties of analytic functions. Math. Z. 61 (1954), 186–199.
Bagemihl, F. and Seidel, W.Regular functions with prescribed measurable boundary values almost everywhere. Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 740–743.
Boivin, A.Carleman approximation on Riemann surfaces. Math. Ann. 275 (1986), no. 1, 57–70.
Boivin, A. and Gauthier, P. M.Holomorphic and harmonic approximation on Riemann surfaces. Approximation, complex analysis, and potential theory (Montreal, QC, 2000), 107–128, NATO Sci. Ser. II Math. Phys. Chem., 37, Kluwer Acad. Publ., Dordrecht, 2001.
Chacrone, S., Gauthier, P. M., and Nersessian, A. H.Carleman approximation on products of Riemann surfaces. Complex Variables, Theory and Application, 37 (1998), no. 1–4, pp 97–111.
Charpentier, Ph.; Dupain, Y.; Mounkaila, M.Approximation par des fonctions holomorphes à croissance controlée. (French) [Approximation by holomorphic functions with controlled growth] Publ. Mat. 38 (1994), no. 2, 269–298.
Collingwood, E. F.; Lohwater, A. J.The theory of cluster sets. Cambridge Tracts in Mathematics and Mathematical Physics, No. 56 Cambridge University Press, Cambridge 1966. [Russian translation, Library of the Journal “Matematika”] Izdat. “Mir”, Moscow, 1971.
Duren, P. L.Theory of\(H^p\)spaces. Pure and Applied Mathematics, Vol. 38 Academic Press, New York-London 1970.
Falcó, J. Gauthier, P. M. Manolaki, M. and Nestoridis, V.A function algebra providing new Mergelyan type theorems in several complex variables. Adv. Math. 381 (2021).
Fornæss, J. E; Forstneriš, F.; Wold, E. F.Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-Weil, and Mergelyan. Advancements in complex analysis—from theory to practice, 133–192, Springer, Cham, [2020], 2020. arXiv: 1802.03924v [math.CV] 12 Feb. 2018.
Garcia, S. R.; Mashreghi, J.; Ross, W. T.Real complex functions. Recent progress on operator theory and approximation in spaces of analytic functions, 91–128, Contemp. Math., 679, Amer. Math. Soc., Providence, RI, 2016.
Gauthier, P.Tangential approximation by entire functions and functions holomorphic in a disc. Izv. Akad. Nauk Armjan. SSR Ser. Mat. 4 1969 no. 5, 319–326.
Gauthier, P. M.Uniform approximation. Complex potential theory (Montreal, PQ, 1993), 235–271, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Kluwer Acad. Publ., Dordrecht, 1994.
Gauthier, P. M. and Hengartner, W.Uniform approximation on closed sets by functions analytic on a Riemann surface. Approximation theory (Proc. Conf. Inst. Math., Adam Mickiewicz Univ., Poznań, 1972), pp. 63–69. Reidel, Dordrecht, 1975.
Gauthier, P. M. and Hengartner, W.Approximation uniforme qualitative sur des ensembles non bornés. Séminaire de Mathématiques Supérieures, 82. Presses de l’Université de Montréal, Montréal, Qué., 1982.
Hakim, M. and Sibony, N., Boundary properties of holomorphic functions in the ball of\(\mathbb C^n\), Math. Ann. 276 (1987), no. 4, 549–555.
Halmos, P. R.Measure Theory. D. Van Nostrand Company, Inc., New York, N. Y., 1950.
Hoffman, K.Banach spaces of analytic functions. Reprint of the 1962 original. Dover Publications, Inc., New York, 1988.
Krantz, S. G. and Min, B.Approximation by holomorphic functions of several complex variables, J. Korean Math. Soc. 57 (2020), No. 5, pp. 1287–1298.
Lehto, O.On the first boundary value problem for functions harmonic in the unit circle. Ann. Acad. Sci. Fenn. Ser. A. I. 1955 (1955), no. 210, 26 pp.
Nersesjan, A. A.Carleman sets. (Russian) Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 6, 465–471.
Noshiro, K.Cluster Sets. Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 28 Springer-Verlag, Berlin-Göttingen-Heidelberg 1960.
Roth, A.Approximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen. Comment. Math. Helv. 11 (1938), 77–125.
Singh, T. B.Introduction to Topology, Springer Nature Singapore Pte Ltd. 2019.
Acknowledgements
The junior author would like to thank the senior author for introducing the problem and for very helpful discussions and warm support throughout the writing of this paper. Furthermore, the junior author is grateful to Professors Dmitry Jakobson and Jacques Hurtubise for the support provided during his postdoctorate at McGill University. We also wish to thank the Fields Institute, in particular, the organizers of the Focus Program on Analytic Function Spaces and their Applications.
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Gauthier, P.M., Shirazi, M. (2023). Radial Limits of Holomorphic Functions in \(\mathbb C^n\) or the Polydisc. In: Binder, I., Kinzebulatov, D., Mashreghi, J. (eds) Function Spaces, Theory and Applications. Fields Institute Communications, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-031-39270-2_9
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