Integral Representation and Time Evolution of Superoscillations

Jussi Behrndt (), Fabrizio Colombo () and Peter Schlosser ()
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Jussi Behrndt: Graz University of Technology, Institute of Applied Mathematics
Fabrizio Colombo: Politecnico di Milano, Dipartimento di Matematica
Peter Schlosser: Graz University of Technology, Institute of Applied Mathematics

Chapter 95 in Operator Theory, 2026, pp 2911-2927 from Springer

Abstract: Abstract Superoscillations are a phenomenon where a band-limited signal locally oscillates faster than its highest frequency component, achieved by combining low-frequency waves to mimic high-frequency oscillations. They are important because they enable super-resolution imaging and play a significant role in weak values in quantum mechanics. The aim of this chapter is to summarize some of the main facts of the mathematics of the theory of superoscillations. Superoscillations will be defined as sequences of holomorphic functions that admit integral representations with respect to complex Borel measures and converge to a plane wave in the space A 1 ( ℂ ) $$\mathcal {A}_1(\mathbb {C})$$ of exponentially bounded entire functions. Additionally, it aims to explain the main results obtained in the study of the evolution of superoscillations via the Schrödinger equation.

Keywords: Superoscillations; Integral representation; Green’s function; Schrödinger equation; 81Q05; 35A20; 35A08 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_100

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