Abstract
In connection with a problem of H. Widom it is shown that if a compact set K on the complex plane contains a smooth Jordan arc on its outer boundary, then the minimal norm of monic polynomials of degree n = 1,2,... is at least (1 + β)cap(K)n with some β > 0, where cap(K)n would be the theoretical lower bound. It is also shown that the rate (1 + o(1))cap(K)n is possible only for compact for which the unbounded component of the complement is simply connected. A related result for sets lying on the real line is also proven.
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Supported by the Europearn Research Council Advanced Grant No. 267055.
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Totik, V. Chebyshev Polynomials on Compact Sets . Potential Anal 40, 511–524 (2014). https://doi.org/10.1007/s11118-013-9357-6
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DOI: https://doi.org/10.1007/s11118-013-9357-6