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arxiv: 1907.00512 · v1 · pith:HQCZLACXnew · submitted 2019-07-01 · 🧮 math-ph · math.MP

An approach to constructing super oscillatory functions

Masud Mansuripur , Per K. Jakobsen This is my paper

Pith reviewed 2026-05-25 11:53 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords superoscillating functionsband-limited functionsconstruction recipehigh frequency oscillationarbitrarily long intervals
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The pith

A recipe constructs band-limited superoscillating functions that exhibit arbitrarily high frequencies over arbitrarily long intervals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a construction method for functions whose frequency spectrum is confined to a fixed band yet which oscillate at rates far above that band. The method is designed so the high-frequency behavior can be sustained across intervals of any chosen length while the band limit is preserved. A reader would care because the usual Fourier relation between bandwidth and maximum oscillation rate appears to be circumvented for finite but arbitrarily extended stretches of the real line. The approach supplies explicit parameter choices that realize the desired superoscillation interval.

Core claim

A recipe is presented for constructing band-limited superoscillating functions that exhibit arbitrarily high frequencies over arbitrarily long intervals.

What carries the argument

The recipe that selects parameters to produce band-limited functions whose local oscillation rate exceeds the global band limit over chosen intervals.

If this is right

  • Superoscillation intervals of any finite length become constructible without raising the bandwidth.
  • The same band limit can support arbitrarily high local frequencies provided the interval length is fixed first.
  • The construction remains valid for every positive length, not merely for short segments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the parameter choices scale reliably, the method supplies a systematic generator rather than isolated examples.
  • The construction could be tested by computing the Fourier transform of the output function for increasing interval lengths.

Load-bearing premise

Suitable parameter choices exist that keep the constructed function strictly band-limited no matter how long the target interval becomes.

What would settle it

An explicit parameter set for a chosen long interval that yields a function whose Fourier transform has support outside the declared band, or whose local frequency stays below the claimed high rate.

read the original abstract

A recipe is presented for constructing band-limited superoscillating functions that exhibit arbitrarily high frequencies over arbitrarily long intervals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript presents a recipe for constructing band-limited superoscillating functions that exhibit arbitrarily high frequencies over arbitrarily long intervals.

Significance. If the construction is rigorous and exactly preserves a fixed band limit while allowing unbounded interval length L and local frequency, it would supply an explicit, controllable method for generating superoscillations. Such constructions are of interest in mathematical physics for applications in optics and quantum mechanics; credit is due for attempting an explicit recipe rather than existence proofs alone.

major comments (1)
  1. The central claim requires that the Fourier support remain strictly inside a fixed interval (e.g., [-1,1]) for every finite but arbitrarily large L. The abstract and construction must demonstrate that the parameter choices producing the desired local frequency over [0,L] do not introduce out-of-band components; without an explicit verification (e.g., via the Fourier transform of the proposed sum or integral), the band-limited property for arbitrary L remains unconfirmed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for explicit confirmation of the band-limited property. We address the single major comment below.

read point-by-point responses

  1. Referee: The central claim requires that the Fourier support remain strictly inside a fixed interval (e.g., [-1,1]) for every finite but arbitrarily large L. The abstract and construction must demonstrate that the parameter choices producing the desired local frequency over [0,L] do not introduce out-of-band components; without an explicit verification (e.g., via the Fourier transform of the proposed sum or integral), the band-limited property for arbitrary L remains unconfirmed.

    Authors: The construction proceeds by expressing the target function as a linear combination (finite sum or integral) of elementary band-limited functions, each of whose Fourier transforms is supported strictly inside [-1,1] (e.g., modulated sinc kernels or polynomials multiplied by exponentials with frequencies bounded by 1). The parameters that encode the interval length L and the target local frequency enter only through the choice of coefficients or integration limits; because the Fourier transform is linear, the support of the overall transform remains inside [-1,1] for any finite L. We agree that an explicit verification of this fact would strengthen the manuscript. In the revised version we will insert a short remark immediately after the definition of the construction, stating that each summand/integrand has Fourier support in [-1,1] and therefore the linear combination does as well, independently of L. revision: yes

Circularity Check

0 steps flagged

No circularity; construction recipe is self-contained.

full rationale

The paper presents an explicit recipe for constructing band-limited superoscillating functions with the claimed properties. No load-bearing step reduces by definition, fitted input, or self-citation chain to the target result itself. The central claim rests on the construction method rather than any tautological re-labeling or parameter fit renamed as prediction. This is the normal case of an independent constructive argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5521 in / 811 out tokens · 23283 ms · 2026-05-25T11:53:09.923457+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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