INTEGRAL REPRESENTATION OF SUPEROSCILLATIONS VIA COMPLEX BOREL MEASURES AND THEIR CONVERGENCE

In the last decade there has been a growing interest in superoscil-lations 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 fea-ture of a superoscillating function in mathematics and quantum mechanics. Also there exists a certain discrepancy between the representation of super-oscillations either as generalized Fourier series, as certain integrals or via spe-cial 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 exist-ing 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.