Abstract
The following principle is well-known in Harmonic Analysis: If a real function has a spectral gap at the origin then it must have many sign changes. We obtain some sharp estimates showing that the set of positivity of such functions cannot be too small. We also extend the principle above to complex functions: If a complex function has a spectral gap at the origin then the variation of argument of this function must be large.
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Ulanovskii, A. The Sturm-Hurwitz Theorem and its Extensions. J Fourier Anal Appl 12, 629–643 (2006). https://doi.org/10.1007/s00041-005-5037-2
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DOI: https://doi.org/10.1007/s00041-005-5037-2