What is Harmonic analysis: Definition and 15 Discussions

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience.
The term "harmonics" originated as the Ancient Greek word harmonikos, meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning.
The classical Fourier transform on Rn is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley–Wiener theorem is an example of this. The Paley–Wiener theorem immediately implies that if f is a nonzero distribution of compact support (these include functions of compact support), then its Fourier transform is never compactly supported (i.e. if a signal is limited in one domain, it is unlimited in the other). This is a very elementary form of an uncertainty principle in a harmonic-analysis setting.
Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis. There are four versions of the Fourier Transform, dependent on the spaces that are mapped by the transformation (discrete/periodic-discrete/periodic: Digital Fourier Transform, continous/periodic-discrete/aperiodic: Fourier Analysis, discrete/aperiodic-continous/periodic: Fourier Synthesis, continous/aperiodic-continous/aperiodic: continous Fourier Transform).

View More On Wikipedia.org
  1. C

    A Compactness and complexity in electrodynamics

    As human beings, we tend to act and observe and think over time periods spanning a few milliseconds to several decades (or even centuries.) Essentially all phenomena that we directly engage with in everyday life are electrodynamical (with quantum electrodynamics over reasonably short time and...
  2. Mr.Husky

    Analysis Opinions on textbooks on Analysis

    What are your opinions on Barry Simon's "A Comprehensive Course in Analysis" 5 volume set. I bought them with huge discount (paperback version). But I am not sure should I go through these books? I have 4 years and can spend 12 hours a week on them. Note- I am now studying real analysis from...
  3. shehanshashika

    Figuring Out the Reactive Energy Measurement Method with Field & Lab Tests

    The digital CT (current transformer) operated energy meters are tested in field using consumer's load conditions by means of a parallel test equipment. It was observed that the reactive energy measurement has high accuracy mismatches on harmonic polluted loads. Some literature indicates that...
  4. jedishrfu

    Elias Stein, Mathematician Of Harmonic Analysis has Died

    The Boston Globe reports that Prof Elias Stein has died at age 87. He was a mathematician who specialized in the mathematics of Harmonic Analysis as applied to other fields such as the stock market and gravitational waves. Prof Terence Tao of UCLA said he had a knack of asking the right...
  5. A

    Relevant highest order harmonics in power systems

    i am learning about harmonic analyzers and i have a couple of questions concerning them that need some sort of practical experience to answer them ( and i have none :) ) : 1-can harmonic analyzer analyze up to any harmonic? or is there a maximum ? if there is no maximum what's the highest...
  6. D

    Modal and Harmonic Analysis of composite

    Can someone help me with the procedures for Modal Analysis and Harmonic Analysis of Composites using ANSYS APDL?
  7. T

    Harmonic Analysis with Ansys APDL: Obtaining Strain and Stress Solutions

    Hi everyone, I have done the Harmonic Analysis of my model with Ansys APDL 17.0 (ACADEMIC version), and I have obtained only the DOF solution of my nodes but i need the Strain and Stress solution . How can I get it? Thanks. Tonino Sepe.
  8. bondking2

    Transition from the second step to the third step?

    Homework Statement How do you change from the second step to the third step? Homework Equations http://i158.photobucket.com/albums/t103/nh4444/Capture_zpspyxdwdna.jpg The Attempt at a Solution My guess is that it has something to do with the geometric series. I am not sure how to proceed.
  9. G

    What is the result of using Euler's equation for Fourier transform integrals?

    when I am using Euler equation for Fourier transform integrals of type \int_{-\infty}^{\infty} dx f(x) exp[ikx] I am getting following integrals: \int_{-\infty}^{\infty} dx f(x) cos(kx) (for the real part) and i* \int_{-\infty}^{\infty} dx f(x) sin(kx) (for its imaginary part) I am...
  10. F

    Application of Lebesgue differentiation theorem

    If ## f\in L_{p}^{\rm loc}(\mathbb{R}^{n}) ## and ## 1\leq p<\infty ##, then a stronger version of Lebesgue differentiation theorem holds: $$\lim\limits_{r\rightarrow 0}\dfrac{\|f\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}{\|\chi_{B(x,r)}\|_{L_{p}(\mathbb{R}^{n})}}=|f(x)|$$ for almost all ##...
  11. salwacheg

    Fatigue Vibration with Fe-Safe

    Hello everybody , I am student in mechanical enginnering. I have to evaluate the solution Fe-Safe. i am working on two subjects: Harmonic and PSD Vibrations. The modal analysis are done in Abaqus. I would like to understand how Fe-Safe computes Life Fatigue in case of harmonic analysis. What...
  12. M

    Questions about harmonic analysis as applied to elasticity

    Hello everyone, I am trying to use a combination of a couple different types of harmonic potential formulations (attached - see question 2) to arrive at an analytic solution to a 3D linear elasticity problem. The problem involves a finite simply connected solid of revolution with mixed...
  13. B

    Ansys tutorials based on harmonic analysis

    can i get ansys tutorials based on harmonic analysis of spring,damper & mass assembly.
  14. K

    Harmonic Analysis: Alloy/Sandwich Panels Theories & Methods

    How can we perform Hramonic Analysis on Sandwich Panels having orthotropic Properties...What are limitation of this analysis...can i have a detailed document related to theortical ,Analytical and Experimental Methods of Perfroming Harmonic (Dynamic Analysis) on Alloy or Sandwich Panels.