Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

85
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
85
Parseval's Theorem for Fourier transform01:15

Parseval's Theorem for Fourier transform

877
Parseval's theorem is a fundamental principle in signal processing that enables the calculation of a signal's energy in either the time domain or the frequency domain. This theorem is pivotal in demonstrating energy conservation between these two domains, ensuring that the computed energy value remains consistent regardless of the domain of analysis.
To understand Parseval's theorem, it is essential to first comprehend how signal energy is typically calculated. When considering a...
877
Convergence of Fourier Series01:21

Convergence of Fourier Series

128
The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
128
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

64
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
64
Trigonometric Fourier series01:17

Trigonometric Fourier series

243
Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...
243
Properties of Fourier series II01:21

Properties of Fourier series II

136
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...
136

You might also read

Related Articles

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

Sort by
Same author

Photoinduced interface reconstruction of Cu<sub>2</sub>O sub-nanocluster-oxygen-rich TiO<sub>2+<i>x</i></sub> heterostructures enabling efficient nitrate-to-ammonia electrosynthesis at industrial current densities.

Chemical science·2026
Same author

FGFR2 Regulates Liver Injury and Repair in a Model of Obstructive Jaundice.

Frontiers in bioscience (Landmark edition)·2026
Same author

A Robust 3D Registration Method via Simultaneous Inlier Identification and Model Estimation.

Journal of imaging·2026
Same author

Liver endothelial zonation orchestrates hepatic steatosis onset through retinoic acid-regulated FGF1.

Science advances·2026
Same author

A noncanonical neuroligin 3-centered complex promotes functional recovery of spinal cord injury.

Stem cell research & therapy·2026
Same author

A systemic review of facial expression recognition (FER) in stroke: diagnosis and emerging applications in rehabilitation.

Frontiers in neurology·2026

Related Experiment Video

Updated: Jun 6, 2025

Network Analysis of Foramen Ovale Electrode Recordings in Drug-resistant Temporal Lobe Epilepsy Patients
09:32

Network Analysis of Foramen Ovale Electrode Recordings in Drug-resistant Temporal Lobe Epilepsy Patients

Published on: December 18, 2016

12.3K

Approximation of functionals on Korobov spaces with Fourier Functional Networks.

Peilin Liu1, Yuqing Liu2, Xiang Zhou3

  • 1School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales 2006, Australia.

Neural Networks : the Official Journal of the International Neural Network Society
|November 28, 2024
PubMed
Summary

This study explores Fourier Functional Networks for learning from functional data, achieving dimension-independent convergence rates. These deep learning models overcome the curse of dimensionality in high-dimensional problems.

Keywords:
Approximation theoryConvolutional neural networkFourier neural operatorKorobov spaceNeural network

More Related Videos

Modeling the Functional Network for Spatial Navigation in the Human Brain
05:55

Modeling the Functional Network for Spatial Navigation in the Human Brain

Published on: October 13, 2023

995
Statistical Modelling of Cortical Connectivity Using Non-invasive Electroencephalograms
08:51

Statistical Modelling of Cortical Connectivity Using Non-invasive Electroencephalograms

Published on: November 1, 2019

5.6K

Related Experiment Videos

Last Updated: Jun 6, 2025

Network Analysis of Foramen Ovale Electrode Recordings in Drug-resistant Temporal Lobe Epilepsy Patients
09:32

Network Analysis of Foramen Ovale Electrode Recordings in Drug-resistant Temporal Lobe Epilepsy Patients

Published on: December 18, 2016

12.3K
Modeling the Functional Network for Spatial Navigation in the Human Brain
05:55

Modeling the Functional Network for Spatial Navigation in the Human Brain

Published on: October 13, 2023

995
Statistical Modelling of Cortical Connectivity Using Non-invasive Electroencephalograms
08:51

Statistical Modelling of Cortical Connectivity Using Non-invasive Electroencephalograms

Published on: November 1, 2019

5.6K

Area of Science:

  • Machine Learning
  • Deep Learning
  • Functional Data Analysis

Background:

  • Deep neural networks are increasingly used for functional data analysis.
  • Theoretical understanding of neural networks for functional data is limited.
  • High-dimensional functional data presents significant challenges.

Purpose of the Study:

  • Investigate the approximation capacity of Fourier Functional Networks.
  • Analyze the theoretical underpinnings of deep learning for functional data.
  • Develop a neural network architecture with reduced parameters for functional data.

Main Methods:

  • Utilized Fourier neural operators and deep convolutional neural networks.
  • Established approximation rates for nonlinear continuous functionals.
  • Focused on functions defined on Korobov spaces of periodic functions.

Main Results:

  • Demonstrated dimension-independent convergence rates for Fourier Functional Networks.
  • Showcased a significant reduction in parameters compared to other architectures.
  • Provided theoretical guarantees for learning from functional data.

Conclusions:

  • Fourier Functional Networks offer a promising approach for high-dimensional functional data analysis.
  • The study overcomes the curse of dimensionality with theoretical insights.
  • The findings pave the way for more efficient and effective deep learning models in this domain.