One cannot hear the winding number
Summary: We construct an example of two continuous maps \(f\) and \(g\) of the circle to itself with the same absolute value of the Fourier transform but with different winding numbers, answering a question of Brézis [cf. \textit{J. Korevaar, Sel. Math., New Ser. 5, No. 1, 107--122 (1999; Zbl 0938.55005)].}
Fourier coefficients, Pauli partners, random Fourier series
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