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In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some convergence to a plane wave is the standard characterizing feature of a superoscillating function in mathematics and quantum mechanics. Also there exists a certain discrepancy between the representation of superoscillations either as generalized Fourier series, as certain integrals or via special functions. The aim of this work is to close these gaps and give a general definition of superoscillations, covering the well-known examples in the existing literature. Superoscillations will be defined as sequences of holomorphic functions, which admit integral representations with respect to complex Borel measures and converge to a plane wave in the space A1(C) of entire functions of exponential type.


This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Transactions of the American Mathematical Society in 2023 following peer review. This article may not exactly replicate the final published version. The definitive publisher-authenticated version is available online at .

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American Mathematical Society



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