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

Parseval's Theorem for Fourier transform01:15

Parseval's Theorem for Fourier transform

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 signal's...
Parseval's Theorem01:18

Parseval's Theorem

Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which expresses a...
Relative Motion Analysis using Rotating Axes01:25

Relative Motion Analysis using Rotating Axes

Consider a component AB undergoing a linear motion. Along with a linear motion, point B also rotates around point A. To comprehend this complex movement, position vectors for both points A and B are established using a stationary reference frame.
However, to express the relative position of point B relative to point A, an additional frame of reference, denoted as x'y', is necessary. This additional frame not only translates but also rotates relative to the fixed frame, making it instrumental in...
IR Spectrum Peak Splitting: Symmetric vs Asymmetric Vibrations01:08

IR Spectrum Peak Splitting: Symmetric vs Asymmetric Vibrations

Identical bonds within a polyatomic group can stretch symmetrically (in-phase) or asymmetrically (out-of-phase). Similar to hydrogen bonding, these vibrations also influence the shape of the IR peak. Generally, asymmetric stretching frequencies are higher than symmetric stretching frequencies. For example, primary amines exhibit two distinct IR peaks between 3300–3500 cm−1 corresponding to the symmetric and asymmetric N-H stretching, while secondary amines exhibit a single stretching vibration...
Wave Parameters01:10

Wave Parameters

The simplest mechanical waves are associated with simple harmonic motion and repeat themselves for several cycles. These simple harmonic waves can be modeled using a combination of sine and cosine functions. Consider a simplified surface water wave that moves across the water's surface. Unlike complex ocean waves, in surface water waves, water moves vertically, oscillating up and down, whereas the disturbance of the wave moves horizontally through the medium. If a seagull is floating on the...
Properties of Fourier series II01:21

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...

You might also read

Related Articles

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

Sort by
Same author

The Association of Hypertriglyceridemic Waist Phenotype with Chronic Kidney Disease and Its Sex Difference: A Cross-Sectional Study in an Urban Chinese Elderly Population.

International journal of environmental research and public health·2016
Same author

Non-Adiabatic Effects on Excited States of Vinylidene Observed with Slow Photoelectron Velocity-Map Imaging.

Journal of the American Chemical Society·2016
Same author

Targeting Heparin to Collagen within Extracellular Matrix Significantly Reduces Thrombogenicity and Improves Endothelialization of Decellularized Tissues.

Biomacromolecules·2016
Same author

Association between sleep duration and the prevalence of hypertension in an elderly rural population of China.

Sleep medicine·2016
Same author

Association between passive smoking and hypertension in Chinese non-smoking elderly women.

Hypertension research : official journal of the Japanese Society of Hypertension·2016
Same author

Morphine versus methylprednisolone or aminophylline for relieving dyspnea in patients with advanced cancer in China: a retrospective study.

SpringerPlus·2016
Same journal

Studying Synchronization of Neural Oscillators through NMDA-AMPA Receptor interactions.

Chaos, solitons, and fractals·2026
Same journal

Prediction of excitable wave dynamics using machine learning.

Chaos, solitons, and fractals·2025
Same journal

A network-based model to assess vaccination strategies for the COVID-19 pandemic by using Bayesian optimization.

Chaos, solitons, and fractals·2025
Same journal

COVID-19 dynamics and immune response: Linking within-host and between-host dynamics.

Chaos, solitons, and fractals·2024
Same journal

Ion gradient-driven bifurcations of a multi-scale neuronal model.

Chaos, solitons, and fractals·2023
Same journal

Growth Feedback Confers Cooperativity in Resource-Competing Synthetic Gene Circuits.

Chaos, solitons, and fractals·2023
See all related articles

Related Experiment Video

Updated: Jun 22, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

Semi-orthogonal Frame Wavelets and Parseval Frame Wavelets Associated with GMRA.

Zhanwei Liu1, Guoen Hu, Guochang Wu

  • 1College of Information Engineering, University of Information Engineering, Zhengzhou, 450002, China.

Chaos, Solitons, and Fractals
|June 4, 2009
PubMed
Summary
This summary is machine-generated.

This study establishes conditions for semi-orthogonal frame wavelets and Parseval frame wavelets (PFWs) in L(2)(R(d)) using matrix dilations. It proves PFWs in generalized multiresolution analysis (GMRA) are equivalent to specific subspaces, offering insights into their properties.

More Related Videos

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Automatic Detection of Highly Organized Theta Oscillations in the Murine EEG
09:35

Automatic Detection of Highly Organized Theta Oscillations in the Murine EEG

Published on: March 10, 2017

Related Experiment Videos

Last Updated: Jun 22, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Automatic Detection of Highly Organized Theta Oscillations in the Murine EEG
09:35

Automatic Detection of Highly Organized Theta Oscillations in the Murine EEG

Published on: March 10, 2017

Area of Science:

  • Harmonic Analysis
  • Functional Analysis
  • Wavelet Theory

Background:

  • Frames and wavelets are crucial in signal processing and mathematical analysis.
  • Matrix dilations generalize standard wavelet constructions.
  • Parseval frame wavelets (PFWs) offer specific desirable properties.

Purpose of the Study:

  • To establish conditions for semi-orthogonal frame wavelets.
  • To characterize Parseval frame wavelets (PFWs) within generalized multiresolution analysis (GMRA).
  • To explore the relationship between PFWs and principal shift invariant spaces.

Main Methods:

  • Analysis of semi-orthogonal and Parseval frame wavelets in L(2)(R(d)).
  • Application of matrix dilations with expanding matrices (A) where |detA| = 2.
  • Investigation of generalized multiresolution analysis (GMRA) and associated subspaces.
  • Utilizing bracket functions and minimal vector-filters for property discovery.

Main Results:

  • A necessary and sufficient condition for a frame wavelet to be semi-orthogonal is derived.
  • A necessary condition for semi-orthogonal frame wavelets is presented.
  • PFWs in GMRA are shown to be equivalent to a specific closed subspace W(0).
  • A property of PFWs associated with GMRA is discovered via their minimal vector-filter.

Conclusions:

  • The study provides a comprehensive framework for understanding semi-orthogonal and Parseval frame wavelets under matrix dilations.
  • The equivalence of PFWs in GMRA to specific subspaces simplifies their analysis.
  • The findings contribute to the theory of wavelets and their applications in advanced mathematical contexts.