Scaling Properties of Superoscillations and the Extension to Periodic Signals

@article{Tang2015ScalingPO,
  title={Scaling Properties of Superoscillations and the Extension to Periodic Signals},
  author={Eugene Tang and Lovneesh Garg and Achim Kempf},
  journal={arXiv: Mathematical Physics},
  year={2015}
}
Superoscillatory wave forms, i.e., waves that locally oscillate faster than their highest Fourier component, possess unusual properties that make them of great interest from quantum mechanics to signal processing. However, the more pronounced the desired superoscillatory behavior is to be, the more difficult it becomes to produce, or even only calculate, such highly fine-tuned wave forms in practice. Here, we investigate how this sensitivity to preparation errors scales for a method for… Expand

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References

SHOWING 1-10 OF 35 REFERENCES
Analysis of superoscillatory wave functions
Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practicalExpand
Unusual properties of superoscillating particles
It has been found that differentiable functions can locally oscillate on length scales that are much smaller than the smallest wavelength contained in their Fourier spectrum—a phenomenon calledExpand
Evolution of quantum superoscillations, and optical superresolution without evanescent waves
A superoscillatory function—that is, a band-limited function f(x) oscillating faster than its fastest Fourier component—is taken to be the initial state of a freely-evolving quantum wavefunction ψ.Expand
Superoscillations: Faster Than the Nyquist Rate
TLDR
This paper investigates the required dynamical range and energy (squared L2 norm) as a function of the superoscillating signals' frequency, number, and maximum derivative, and shows that the required energy grows exponentially with the number ofsuperoscillations, and polynomially with the reciprocal of the bandwidth. Expand
Direct Construction of Superoscillations
  • D. Lee, P. Ferreira
  • Mathematics, Computer Science
  • IEEE Transactions on Signal Processing
  • 2014
TLDR
This work considers signals of fixed bandwidth and with a finite or infinite number of samples at the Nyquist rate, which are regarded as the adjustable signal parameters and shows that this class of signals can be made to superoscillate by prescribing its values on an arbitrarily fine and possibly nonuniform grid. Expand
Superoscillations with Optimal Numerical Stability
TLDR
Time translation σ is introduced as a design parameter and an explicit closed formula is given for the condition number of the matrix of the problem, as a function of σ, which enables the best possible condition number to be determined. Expand
Temporal Pulse Compression Beyond the Fourier Transform Limit
It is a generally known that the Fourier transform limit forbids a function and its Fourier transform to both be sharply localized. Thus, this limit sets a lower bound to the degree to which aExpand
Evanescent and real waves in quantum billiards and Gaussian beams
A mode in a confined planar region can contain evanescent waves in its plane-wave superposition. It would seem impossible to construct such a mode by continuation of an external scatteringExpand
Black Holes, Bandwidths and Beethoven
It is usually believed that a function φ(t) whose Fourier spectrum is bounded can vary at most as fast as its highest frequency component ωmax. This is, in fact, not the case, as Aharonov, Berry, andExpand
Superoscillations with arbitrary polynomial shape
We present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval. These functionsExpand
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