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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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 , $f(0)=0$, $f(1)=0$...