# 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.

View More On Wikipedia.org

2. ### 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...

8. ### 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...
9. ### 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...
10. ### 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...
11. ### 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...
12. ### 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...
13. ### 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...
14. ### 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...
15. ### 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 ##...
16. ### 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...
17. ### "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.
18. ### 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...
19. ### 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...
20. ### 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...
21. ### 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...
22. ### 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...
23. ### Fourier transform of rectangular pulse

Here is the question: Here is my answer