Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Oscillations about an Equilibrium Position

Oscillations about an Equilibrium Position

Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so because...

Damped Oscillations

Damped Oscillations

In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...

Properties of Fourier series II

Properties of Fourier series II

Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
A function f(t) is...

The Squeeze Theorem

The Squeeze Theorem

Certain mathematical functions exhibit unpredictable or highly variable behavior near specific input values, making direct evaluation of their limits challenging. This complexity may arise from rapid oscillations or irregular patterns that obscure the function’s trend. In such cases, the Squeeze Theorem offers a reliable method for determining limits.According to the Squeeze Theorem, if a function is confined between two other functions near a particular point, and both outer functions approach...

Bessel Function of Order Zero

Bessel Function of Order Zero

A common physical example of wave propagation with radial symmetry is the ripple formed when a stone is dropped into a still pond. The disturbance originates at a central point and travels outward as a circular wave. As the radius of the wavefront increases, the same initial energy is distributed along a progressively larger circumference. Consequently, the amplitude, or height, of the wave decreases with distance from the center. This decay behavior cannot be captured by simple sine or cosine...

Limits with Oscillating Discontinuities

Limits with Oscillating Discontinuities

An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the most...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

IMPROVEMENT OF COGNITIVE FUNCTION IN WISTAR RATS UNDER CHRONIC STRESS CONDITIONS WITH MELATONIN.

Georgian medical news·2025

Same author

A new concept for temporal gating of synchrotron X-ray pulses.

Journal of synchrotron radiation·2021

Same author

The preoperative management of Hepatitis C may improve the outcome after total knee arthroplasty.

The bone & joint journal·2019

Same author

Can some early revision total hip arthroplasties be avoided?

The bone & joint journal·2019

Same author

Sol-gel Synthesis, Photo- and Electrocatalytic Properties of Mesoporous TiO<sub>2</sub> Modified with Transition Metal Ions.

Nanoscale research letters·2017

Same author

Joint analysis of BICEP2/keck array and Planck Data.

Physical review letters·2015

Search research articles

Related Experiment Video

Updated: Jul 20, 2026

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Published on: May 30, 2014

Oscillation of functions with a spectral gap.

A Eremenko¹, D Novikov

¹Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. eremenko@math.purdue.edu

Proceedings of the National Academy of Sciences of the United States of America

|April 14, 2004

Summary

We proved a conjecture about function oscillation, showing that functions with a spectral gap at the origin have a minimum sign change density. This finding is crucial for understanding function behavior and spectral analysis.

More Related Videos

A Microfluidics Approach for the Functional Investigation of Signaling Oscillations Governing Somitogenesis

A Microfluidics Approach for the Functional Investigation of Signaling Oscillations Governing Somitogenesis

Published on: March 19, 2021

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

Related Experiment Videos

Last Updated: Jul 20, 2026

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Published on: May 30, 2014

A Microfluidics Approach for the Functional Investigation of Signaling Oscillations Governing Somitogenesis

A Microfluidics Approach for the Functional Investigation of Signaling Oscillations Governing Somitogenesis

Published on: March 19, 2021

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

Area of Science:

Real analysis
Fourier analysis
Measure theory

Background:

Oscillation of functions is a key property studied in mathematical analysis.
Understanding the behavior of functions with spectral gaps is important for various applications.
Previous conjectures regarding the oscillation of such functions remained unproven.

Purpose of the Study:

To prove a long-standing conjecture concerning the oscillation of functions with a spectral gap at the origin.
To establish a lower bound for the asymptotic density of sign changes for a specific class of functions.

Main Methods:

The study involves analyzing the Fourier transform of real measures on the real line.
The core of the method relies on the property that the Fourier transform is zero within a specific interval (-a, a).
Asymptotic analysis as r approaches infinity is employed to determine the density of sign changes.

Main Results:

It is proven that for a real measure f, if its Fourier transform is zero on (-a, a), the asymptotic lower density of its sign changes on [0, r) is at least a/pi as r approaches infinity.
This result holds for measures with a certain growth rate at infinity.
The statement is shown to be false for measures that grow faster than a specific threshold.

Conclusions:

The conjecture on the oscillation of functions with a spectral gap at the origin is proven.
A precise lower bound for the sign change density is established, dependent on the spectral gap parameter 'a'.
The findings highlight the critical role of the measure's growth rate at infinity in determining its oscillation properties.