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<p>Superoscillations have roots in various scientific disciplines, including optics, signal processing, radar theory, and quantum mechanics. This intriguing mathematical phenomenon permits specific functions to oscillate at a rate surpassing their highest Fourier component. A different way of thinking about superoscillations consists in realizing that it is possible to reproduce the exponential function far away from the origin by only knowing its value in a countable set of points near the origin. By using this perspective, one can extend the idea of superoscillations to functions that are not a sum of exponential functions, namely to the notion of supershift. The study of time evolution for superoscillations (and supershifts) has naturally lead to their extension to the case of several variables. In this paper, however, we take a different approach and use the theory of Bernstein and Lagrange approximation of analytic functions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {C}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> to obtain deeper results for the several variables case. We provide specific examples related to harmonic analysis where the variables vary in multi-dimensional frequency (space, or scale) domains.</p>

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Analyticity, superoscillations and supershifts in several variables

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月14日

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  • F. Colombo,
  • I. Sabadini,
  • D. C. Struppa,
  • A. Yger

摘要

Superoscillations have roots in various scientific disciplines, including optics, signal processing, radar theory, and quantum mechanics. This intriguing mathematical phenomenon permits specific functions to oscillate at a rate surpassing their highest Fourier component. A different way of thinking about superoscillations consists in realizing that it is possible to reproduce the exponential function far away from the origin by only knowing its value in a countable set of points near the origin. By using this perspective, one can extend the idea of superoscillations to functions that are not a sum of exponential functions, namely to the notion of supershift. The study of time evolution for superoscillations (and supershifts) has naturally lead to their extension to the case of several variables. In this paper, however, we take a different approach and use the theory of Bernstein and Lagrange approximation of analytic functions in \({\mathbb {C}}^n\) C n to obtain deeper results for the several variables case. We provide specific examples related to harmonic analysis where the variables vary in multi-dimensional frequency (space, or scale) domains.