What is Fourier: Definition and 1000 Discussions

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.
The Fourier transform of a function of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation, as proven by the Fourier inversion theorem.

Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint. The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or Rn (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.

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  1. kakaho345

    Finding free electron gas Green function in Fourier space

    As in title: Plugging in the definition is straight forward, I am too lazy to type, I will just quote the book Fetter 1971: Up to here everything is very straight forward, in particular, since we are working on free electron gas, ##E=\hbar \omega## However, I have no idea how to arrive...
  2. T

    I Fourier Transform of Photon Emission Hamiltonian

    Hey all, I just wanted to double check my logic behind getting the Fourier Transform of the following Hamiltonian: $$H(x) = \frac{ie\hbar}{mc}A(x)\cdot\nabla_{x}$$ where $$A(x) = \sqrt{\frac{2\pi\hbar c^2}{\omega L^3}}\left(a_{p}\epsilon_{p} e^{i(p\cdot x)} + a_{p}^{\dagger}\epsilon_{p}...
  3. Euge

    POTW Fourier Series on the Unit Interval

    Evaluate the Fourier series $$\frac{1}{\pi^2}\sum_{k = 1}^\infty \frac{\cos 2\pi kx}{k^2}$$ for ##0 \le x \le 1##.
  4. Skaiserollz89

    A Fourier optics model of a 4f system

    In my system I am trying to represent two lenses. L1 with focal length f1=910mm and the other lens, L2 with focal length f2=40mm. These lenses are space such that there is a distance of f1+f2 between the lenses. I have a unit amplitude plane wave incident on L1. My goal is to find the...
  5. nomadreid

    I Is this the Fourier Number, or some other formula?

    This is on a building in Korea, and the F0 made me think of the Fourier number, but the rest of the formula is unfamiliar to me. Does anyone recognize it?
  6. redtree

    I Properties of the Fourier transform

    I was wondering if the following is true and if not, why? $$ \begin{split} \hat{f}_1(\vec{k}) \hat{f}_2(\vec{k}) &= \hat{f}_1(\vec{k}) \int_{\mathbb{R}^n} f_2(\vec{x}) e^{-2 \pi i \vec{k} \cdot \vec{x}} d\vec{x} \\ &= \int_{\mathbb{R}^n} \hat{f}_1(\vec{k}) f_2(\vec{x})...
  7. S

    Fourier transform of ##e^{-a |t|}\cos{(bt)}##

    First, ##\tilde{f}(\omega)=\int_{-\infty}^{\infty}e^{a|t|}\cos(bt)e^{-i\omega t} \mathrm{d}t## We can get rid of the absolute value by splitting the integral up ##\int_{-\infty}^{0}e^{at}\cos(bt)e^{-i\omega t} \mathrm{d}t+ \int_{0}^{\infty}e^{-at}\cos(bt)e^{-i\omega t} \mathrm{d}t## Using...
  8. M

    Lorentzian line profile of emitted radiation

    First of all i tried to follow the textbook. Here they start of by modelling the atom as an harmonic oscilator: Then they find the solution as: They neglect the second term as omega_0 >> gamma which also makes good sense so they end up with: So far so good. After this they state the...
  9. chwala

    Solving the Fourier cosine series

    My question is; is showing the highlighted step necessary? given the fact that ##\sin (nπ)=0##? My question is in general i.e when solving such questions do i have to bother with showing the highlighted part... secondly, Can i have ##f(x)## in place of ##x^2##? Generally, on problems to do...
  10. P

    Python Floating point arithmetic and Fourier collocation

    >>> from numpy import exp, pi >>> exp(1j*pi) (-1+1.2246467991473532e-16j) The fact that the imaginary part of this is not zero is wrecking a fourier collocation scheme for a nonlinear PDE with periodic boundary conditions: the coefficient corresponding to the Nyquist frequency, which should be...
  11. P

    V Space With Norm $||*||$ - Fourier Series

    Hi, a question regarding something I could not really understand The question is: Let V be a space with Norm $||*||$ Prove if $v_n$ converges to vector $v$. and if $v_n$ converges to vector $w$ so $v=w$ and show it by defintion. The question is simple, the thing I dont understand, what...
  12. S

    Fourier transform of wave packet

    I am unsure if ##h(x,t)## really is a wave packet, but it looks like one, hence the title. Anyway, so I'd like to determine ##\hat{h}(k,t=0)##. My attempt so far is recognizing that, without the real part in the integral, i.e. ##g(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} a(k)e^{i(kx-\omega...
  13. S

    Comp Sci Plot periodic function with Fourier coefficients

    I have plotted the function for ##T=15## and ##\tau=T/30## below with the following code in Python: import numpy as np import matplotlib.pyplot as plt def p(t,T,tau): n=np.floor(t/T) t=t-n*T if t<(2*np.pi*tau): p=np.sin(t/tau) else: p=0 return p...
  14. S

    Characterize Fourier coefficients

    I would try to determine whether ##p(t)## is even or odd. This would be so much easier if the values of ##\tau## and ##T## would be specified, but maybe it's possible to do without it, which I'd prefer. If for example ##\tau=1/2## and ##T=2\pi##, then ##p(t)=\sin{(2t)}## for ##0\leq t <\pi ##...
  15. H

    A Polar Fourier transform of derivatives

    The 2D Fourier transform is given by: \hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy In terms of polar co-ordinates: \hat{f}(\rho,\phi)=\int_{0}^{\infty}\int_{-\pi}^{\pi}rf(r,\theta)e^{-ir\rho\cos(\theta-\phi)}drd\theta For Fourier transforms in cartesian co-ordinates, relating the...
  16. S

    "Simple" Fourier transform problem

    I am unsure about what is being asked for in the question. At first I thought the question asks one to calculate the inverse Fourier transform and then to analyze its depends on ##t##, however, the "estimate" makes me think otherwise.
  17. A

    Fourier sine and cosine transforms of Heaviside function

    Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. I tried each part of the problem, but I'm only left in terms of limits as x -> infinity of sin or cos...
  18. S

    Simple Fourier transformation calculation

    So, ##\hat{p}(\omega)=\int_{-\infty}^{\infty} p(t)e^{-i\omega t}\mathrm{d}t=A\int_{0}^{\infty}e^{-t(\gamma+i(\omega+\omega_0))}=A\left[-\frac{e^{-t(\gamma+i(\omega+\omega_0))}}{\gamma+i(\omega+\omega_0)}\right]_0^\infty,## provided ##\gamma+ i(\omega+\omega_0)\neq 0## for the last equality. Now...
  19. Salmone

    I Doubts about Fourier transform of IR spectroscopy

    I was studying a Michelson interferometer for infrared absorption in Fourier transform and I've found these two images (taken from https://pages.mtu.edu/~scarn/teaching/GE4250/ftir_lecture_slides.pdf ) representing an infrared monochromatic beam of light going into the interferometer and the...
  20. QuarkDecay

    Fourier transformation for circular apertures

    My notes say that the Resolution of the Aperture(in the Electric field of the wave) is the Fourier transformation of the aperture. Then gives us the equation of the aperture: and says that for the circular aperture in particular also: My attempt at solving this: We know that the Fourier...
  21. K

    A Fourier Transform MW spectroscopy in a FB cavity

    Hello! I am reading about Fourier Transform MW spectroscopy in a FB cavity, which seems to be quite an old technique and I want to make sure I got it right. As far as I understand, this is very similar to normal NRM, i.e. one applies a MW ##\pi/2## pulse which puts the molecules in a linear...
  22. Salmone

    I Fourier transform of a beat

    What is the Fourier transform of a beat? For example, I want to calculate the Fourier transform of the function ##f(t)=\cos((\omega_p+\omega_v) t)+\cos((\omega_p-\omega_v)t),## where ##$\omega_p+\omega_v=\Omega,\space\omega_p-\omega_v=\omega## and ##\Omega\simeq\omega.## I think it is equal to...
  23. M

    I Please discuss discrete Fourier analysis

    It has been 35 years since I did the math for Fourier analysis, and I have forgotten what the subtleties are. Please be kind. So this is not a how do I calculate a DFT (though that may be my next question) but rather how do I use it, and interpret the results. All the online and software I find...
  24. F

    Fourier transform: duality property and convolution

    Hello, First of all, I checked several other threads mentioning duality, but could not find a satisfying answer, and I don't want to revive years old posts on the subject; if this is bad practice, please notify me (my apologies if that is the case). For the past few days, I have had a lot of...
  25. D

    Fourier transform ##f(t) = te^{-at}##

    Doing the Fourier transform for the function above I'm getting a result, but since I can't get the function f(t) with the inverse Fourier transform, I'm wondering where I made a mistake. ##F(w) = \frac{1}{\sqrt{2 \pi}} \int_{0}^{\infty} te^{-t(a + iw)} dt## By integrating by part, where G = -a...
  26. D

    B Fourier Analysis on musical chords in different instruments

    I wanted to do an investigation about how the same musical chord can have the same pitch but sound different on different musical instruments. Like how chord C major would sound higher played in the electric guitar than a C major played on piano. How should I approach this investigation?
  27. warhammer

    Fourier Transforms -- Please check my solution

    My Professor has started on the Fourier Transforms Topic in the Introductory Mathematical Physics class and gave us a small homework to try our concepts on. I have attached a clear & legible snippet of my solution. I request someone to please have a look at it & determine if my solution is...
  28. Swordwhale

    Fourier transform radial component of magnetic field

    Hello everybody! I have a question concerning the Fourier transformation: So far I have experimentially measured the magnetic field of a quadrupole but as the hall effect sensor had a fixed orientation I did two series, one for the x, one for y component of the magnetic field, I have 50 values...
  29. C

    I What's the definition of "periodic extension of a function"?

    I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...
  30. C

    A Calculation of Fourier coefficients using SAMBA methodology

    Hello everyone. I have 4 samples of 50 elements from 4 unknown random variables obtained from a Karhunen-Loève decomposition using Matlab's pca (each one is a column of size 50 from the coefficient matrix). I am following the article SAMBA: Sparse Approximation of Moment-Based Arbitrary...
  31. R

    Fourier series, periodic function for a string free at each end

    From the statement above, since the ring is massless, there's no force acting vertically on the rings. Thus, the slope is null. ##\frac{\partial y(0,0)}{\partial x} = \frac{\partial y(L,0)}{\partial x} = 0## ##\frac{\partial y(0,0)}{\partial x} = A\frac{2 \pi}{L}cos(\frac{2 \pi 0}{L}) =...
  32. G

    How do dark spot and tiny hole filters affect Fourier optics images?

    I was assigned an experiment of Fouriers optics where I have to use different Filters. One of them was the dark spot and the tiny hole. As of my understanding, for tiny hole, we cut off all high-frequency light related to diffraction and refraction, thus using only the low freuency part of the...
  33. L

    Fourier transform to solve this Laplace equation

    I have tried to Fourier transform in ##x## and get the result in the transformed coordinates, please check my result: $$ \tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky} $$ However, I'm having some problems with the inverse transform: $$ \frac{1}{2\pi}\int_{-\infty}^\infty...
  34. LCSphysicist

    How do I know "what Fourier transform" to use?

    Homework Statement:: . Relevant Equations:: . I am having a hard time thinking about Fourier transform, because there are so many conventions that i think i got more confused each time i think about it. See an example, "Find the Fourier transform of $$V(t) = Ve^{iwt} \text{ if } nT \leq t...
  35. S

    MHB Fourier Transform Help: Issues Solving for a & b

    Hello again. Having some issues on Fourier transform. Can someone please tell me how to proceed? Need to solve this then use some software to check my answer but how to solve for a and b. Plzz help
  36. redtree

    I Fourier transform of a function in spherical coordinates

    I am trying to understand the relationship between Fourier conjugates in the spherical basis. Thus for two functions ##f(\vec{x}_3)## and ##\hat{f}(\vec{k}_3)##, where \begin{equation} \begin{split} \hat{f}(\vec{k}_3) &= \int_{\mathbb{R}^3} e^{-2 \pi i \vec{k}_3 \cdot \vec{x}_3} f(\vec{x}_3...
  37. tworitdash

    A Fourier Transform of an exponential function with sine modulation

    I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time. The time-domain form is, s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}} Here, \mu , \Omega and \lambda are constants. A quick...
  38. S

    MHB Fourier Series involving Hyperbolic Functions

    Hello everyone first time here. don't know if it's the correct group... Am having some issues wiz my maths homework that going to count as a final assessment. Really Really need help. The function (f), with a period of 2π is : f(x) = cosh(x-2π) if x [π;3π].. I had to do a graph as the first...
  39. A

    The value of a Fourier series at a jump point (discontinuity)

    Greetings according to the function we can see that there is a jump at x=e and I know that the value of the function at x=e should be the average between the value of f(x) at this points my problem is the following the limit of f(x) at x=e is -infinity and f(e)=1 how can we deal with such...
  40. mtv65

    A S-duality in Physics and Fourier Transforms

    In many articles, authors compare S-duality in physics to Fourier transforms. For example: Joseph Polchinski, in his article "String Duality" (hep-th/9607050v2), writes "Weak/strong duality [...] is similar to a Fourier transform, where a function which becomes spread out in position space...
  41. H

    Fourier transform to solve PDE (2nd order)

    I just want to make sure I am on the right track here (hence have not given the other information in the question). In taking the Fourier transform of the PDE above, I get: F{uxx} = iω^2*F{u}, F{uxt} = d/dt F{ux} = iω d/dt F{u} F{utt} = d^2/dt^2 F{u} Together the transformed PDE gives a second...
  42. F

    A Fourier transform and Cosmic variance - a few precisions

    I cite an original report of a colleague : 1) I can't manage to proove that the statistical error is formulated like : ##\dfrac{\sigma (P (k))}{P(k)} = \sqrt{\dfrac {2}{N_{k} -1}}_{\text{with}} N_{k} \approx 4\pi \left(\dfrac{k}{dk}\right)^{2}## and why it is considered like a relative error ...
  43. docnet

    I The precise relationship between Fourier series and Fourier transform

    Would someone be able to explain like I am five years old, what is the precise relationship between Fourier series and Fourier transform? Could someone maybe offer a concrete example that clearly illustrates the relationship between the two? I found an old thread that discusses this, but I...
  44. docnet

    Find the Fourier Coefficients

    Consider the function ##f:[0,1]\rightarrow \mathbb{R}## given by $$f(x)=x^2$$ (1) The Fourier coefficients of ##f## are given by $$\hat{f}(0)=\int^1_0x^2dx=\Big[\frac{x^3}{3}\Big]^1_0=\frac{1}{3}$$ $$\hat{f}(k)=\int^1_0x^2e^{-2\pi i k x}dx$$ Can this second integral be evaluated?
  45. U

    Help with evaluating this Fourier transform

    The definition of Fourier transform (F.T.) that I am using is given as: $$f(\vec{x},t)=\frac{1}{\sqrt{2\pi}}\int e^{-i\omega t}\tilde{f}(\vec{x},\omega)\,\mathrm{d}\omega$$ I want to show that: $$\frac{1}{c\sqrt{2\pi}}\int e^{-i\omega t}\omega^2...
  46. A

    I How can one see the noise in the Fourier Domain (Nyquist Frequency)?

    Say we have a transform of a line profile that extends out to the Nyquist frequency such that you cannot see the noise level, what could you change in your spectrograph arrangement that would allow you to see the noise level in the Fourier domain? My thought is that we can apply a filter, P(s)...