What is Fourier decomposition: Definition and 13 Discussions

In mathematics, a Fourier series () is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

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  1. Wrichik Basu

    B Clarification of Notation - Fourier decomposition of fields in QFT

    I am studying QFT from A First Book of QFT. It is a very well-written book. However, due to some personal reasons, I cannot buy the printed book at this moment. So I borrowed this book from a person (who, in turn, borrowed it from his university library), and scanned it. Everything is fine...
  2. R

    Fourier Series of Sawtooth Wave from Inverse FT

    Homework Statement I want to find the Fourier series of the sawtooth function in terms of real sine and cosine functions by using the formula: $$f_p (t)=\sum^\infty_{k=-\infty} c_k \exp \left(j2\pi \frac{k}{T}t \right) \tag{1}$$ This gives the Fourier series of a periodic function, with the...
  3. E

    I Decomposing a Function for Numerical Integration

    Is their a tutorial or a reference on how to decompose a function, specifically Fourier and Legendre decomposition, for numerical integration? The method I am going to use for the numerical integration is the Gauss Quadrature, and I suppose I need to decompose my function for the rule to work...
  4. D

    A A question about the mode expansion of a free scalar field

    In the canonical quantisation of a free scalar field ##\phi## one typical constructs a mode expansion of the corresponding field operator ##\hat{\phi}## as a solution to the Klein-Gordon equation...
  5. Ma Xie Er

    A Why Is the Equality in This Spectral Analysis Proof Correct?

    I'm reading "Time Series Analysis and Its Applications with R examples", 3rd edition, by Shumway and Stoffer, and I don't really understand a proof. This is not for homework, just my own edification. It goes like this: Σt=1n cos2(2πtj/n) = ¼ ∑t=1n (e2πitj/n - e2πitj/n)2 = ¼∑t=1ne4πtj/n + 1 + 1...
  6. F

    I Solution to PDEs via Fourier transform

    Suppose a PDE for a function of that depends on position, ##\mathbf{x}## and time, ##t##, for example the wave equation $$\nabla^{2}u(\mathbf{x},t)=\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)$$ If I wanted to solve such an equation via a Fourier transform, can I Fourier...
  7. E

    Force on wheels in an accelerating vehicle

    Assume that a motorcycle of mass m has two wheels that are equidistant from its centre i.e the force on each wheel is m*g/2. If the motorcycle accelerates forward, will the two forces on each wheel (measured instantaneously) remain the same? If not, how can one mathematically describe the...
  8. G

    How Fourier components of vector potential becomes operators

    Hello. I'm studying quantization of electromagnetic field (to see photon!) and on the way to reach harmonic oscillator Hamiltonian as a final stage, sudden transition that the Fourier components of vector potential A become quantum operators is observed. (See...
  9. J

    What Is the True Nature of Electrons According to de Broglie's Theory?

    This is going to seem like it should be an educational question, and perhaps it should, but please bear with me, because I think that it has theoretical content. A nice quote that I recently heard someplace went something like: "We used to argue about whether electrons [etc] were particles or...
  10. R

    What Does the Fourier Decomposition of an Image Represent?

    Hello everyone have a look at this video of Fourier Decomposition of an image.also we know that Fourier series is given in the image as...
  11. D

    Electrostatics Fourier Decomposition (problem setting up boundaries)

    Homework Statement An #a*b*c box is given in x,y,z (so it's length #a along the x axis, etc.). Every face is kept at #V=0 except for the face at #x=a , which is kept at #V(a,y,z)=V_o*sin(pi*y/b)*sin(pi*z/c). We are to, "solve for all possible configurations of the box's potential" Homework...
  12. H

    Nightclub physics: Fourier decomposition with cocktail glasses

    This is purely conceptual and I'm just looking for opinions on whether its misguided or, indeed, plausible. From what I understand about Fourier decomposition we can break down an analog signal into component sinusoidal waves. My thinking is that the sound system at a nightclub can be...
  13. W

    Fourier decomposition and heat equation

    Homework Statement In the heat equation, we have $T(t,x)=sum of a_k(t)b_k(x)$. Now I want to find a formula for computing the initial coefficients $a_k(0)$ given the initial temperature distribution $f(x)$. Homework Equations We know that in a heat equation , $f(0)=0$, $f(1)=0$...
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