We give a negative answer to the Newman–Shapiro problem on weighted approximation for entire functions formulated in 1966 and motivated by the theory of operators on the Fock space. There exists a function in the Fock space such that its exponential multiples do not approximate some entire multiples in the space. Furthermore, we establish several positive results under different restrictions on the function in question.
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Yurii Belov, Alexander Borichev, The Newman–Shapiro problem. J. Eur. Math. Soc. 25 (2023), no. 4, pp. 1227–1251DOI 10.4171/JEMS/1209