Integral representation of superoscillations via complex Borel measures and their convergence

In the last decade there has been a growing interest in superoscillations various fields of mathematics, physics and engineering. However, while applications as optics local oscillatory behaviour is important property, some convergence to plane wave standard characterizing feature superoscillating function mathematics quantum mechanics. Also exists certain discrepancy between representation either generalized Fourier series, integrals or via special functions. The aim this work close these gaps give general definition superoscillations, covering well-known examples existing literature. Superoscillations will be defined sequences holomorphic functions, which admit integral representations with respect complex Borel measures converge space A 1 ( mathvariant="double-struck">C stretchy="false">) \mathcal {A}_1(\mathbb {C}) entire functions exponential type.