Abstract
This is an expository paper on a topic of classical analysis arising from the BMO-theory of topological degree (Brezis-Nirenberg, 1995). We sketch the history of the subject and some of its recent developments.
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References
J. Bourgain and J.-P. Kahane, “Sur les séries de Fourier des fonctions continues unimodulaires,” Ann. Inst. Fourier 60(4), 1201–1214 (2010).
J. Bourgain and G. Kozma, “One Cannot Hear the Winding Number,” J. Eur. Math. Soc. 9, 637–658 (2007).
H. Brezis, “New Questions Related to the Topological Degree,” in The Unity of Mathematics: In Honour of the 90th Birthday of I.M. Gelfand (Birkhäuser, Boston, 2006), Prog. Math. 244, pp. 137–154.
H. Brezis and J.-M. Coron, “Large Solutions for Harmonic Maps in Two Dimensions,” Commun. Math. Phys. 92, 203–215 (1983).
H. Brezis and J. Nirenberg, “Degree Theory and BMO. I: Compact Manifolds without Boundaries,” Sel. Math. 1, 197–263 (1995).
G. H. Hardy, Divergent Series (Clarendon Press, Oxford, 1949; Chelsea, New York, 1991).
J.-P. Kahane, “Sur l’équation fonctionnelle ∫π(ϕ(t + s) − ϕ(s))3 ds = sint,” C. R., Math., Acad. Sci. Paris 341, 141–145 (2005).
J.-P. Kahane, “Winding Numbers and Summation Processes,” Complex Var. Elliptic Eqns. 55, 911–922 (2010).
J. Korevaar, “On a Question of Brézis and Nirenberg Concerning the Degree of Circle Maps,” Sel. Math. 5, 107–122 (1999).
B. Kuttner, “The Relation between Riemann and Cesàro Summability,” Proc. London Math. Soc., Ser. 2, 38, 273–283 (1935).
A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959), Vol. 1.
A. Zygmund, “Sur la dérivation des séries de Fourier,” Bull. Acad. Pol. A, 243–249 (1924).
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Kahane, JP. Winding numbers and fourier series. Proc. Steklov Inst. Math. 273, 191–195 (2011). https://doi.org/10.1134/S0081543811040080
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DOI: https://doi.org/10.1134/S0081543811040080