Abstract
For a measure μ on the complex plane μ-regular points play an important role in various polynomial inequalities. In the present work it is shown that every point in the set {μ′>0} (actually of a larger set where μ is strong) with the exception of a set of zero logarithmic capacity is a μ-regular point. Here “set of zero logarithmic capacity” cannot be replaced by “β-logarithmic Hausdorff measure 0” with β=1 (it can be replaced by “β-logarithmic measure 0” with any β>1). On the other hand, for arbitrary μ the set of μ-regular points can be quite small, but never empty.
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References
A. Ancona, Démonstration d’une conjecture sur la capacité et l’effilement, C. R. Acad. Sci. Paris, 297 (1983), 393–395.
E. Levin and D. S. Lubinsky, The size of the set of μ-irregular points of a measure μ, Acta Math. Hungar., 133 (2011), 242–250.
T. Ransford, Potential Theory in the Complex plane, Cambridge University Press (Cambridge, 1995).
E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften, 316, Springer-Verlag (New York/Berlin, 1997).
H. Stahl and V. Totik, General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications, 43, Cambridge University Press (Cambridge, 1992).
M. Tsuji, Potential Theory in Modern Function Theory, Maruzen (Tokyo, 1959).
J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 3rd ed., Amer. Math. Soc. Colloquium Publications, XX, Amer. Math. Soc. (Providence, 1960).
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Totik, V. The size of irregular points for a measure. Acta Math Hung 136, 222–231 (2012). https://doi.org/10.1007/s10474-011-0177-0
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DOI: https://doi.org/10.1007/s10474-011-0177-0