Questions tagged [fourier-analysis]

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

Fourier analysis is the study of how general functions can be decomposed into trigonometric or exponential functions with definite frequencies. There are two types of Fourier expansions:

  • Fourier series: If a (reasonably well-behaved) function is periodic, then it can be written as a discrete sum of trigonometric or exponential functions with specific frequencies.
  • Fourier transform: A general function that isn’t necessarily periodic (but that is still reasonably well-behaved) can be written as a continuous integral of trigonometric or exponential functions with a continuum of possible frequencies.

The reason why Fourier analysis is so important is that many (although certainly not all) of the differential equations that govern physical systems are linear, which implies that the sum of two solutions is again a solution. Therefore, since Fourier analysis tells us that any function can be written in terms of sinusoidal functions, we can limit our attention to these functions when solving the differential equations. And then we can build up any other function from these special ones. This is a very helpful strategy, because it is invariably easier to deal with sinusoidal functions than general ones.

Fourier series

Consider a function $f(x)$ that is periodic on the interval $0 ≤ x ≤ L$, then Fourier’s theorem states that $f(x)$ can be written as $$f(x)={a_0}+\sum_{n=1}^{\infty}[a_n \cos (\frac{2n\pi x}{L})+b_n \sin (\frac{2n\pi x}{L})]$$ where the constant coefficients $a_n$ and $b_n$ are called the Fourier coefficients of $f$ and is given by $$a_0=\frac{1}{L}\int_0^L f(x) dx$$ $$a_n=\frac{2}{L}\int_0^L f(x)\cos (\frac{2\pi nx }{L}) dx$$ $$b_n=\frac{2}{L}\int_0^L f(x)\sin (\frac{2\pi nx }{L}) dx$$


Fourier Transform:

For this part find the following link

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Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I believe many of you…
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Fourier transform for dummies

What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on the question of Kevin Lin, which didn't quite fit in Mathoverflow. Answers at any level of sophistication are…
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What is the difference between Fourier series and Fourier transformation?

What's the difference between Fourier transformations and Fourier Series? Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?
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Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. What is the connection between these two? Is there a way to get from one to the other (and back again)? Is there an overall, connecting (geometric?)…
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Difference between Fourier transform and Wavelets

While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in…
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Fourier Transform of Derivative

Consider a function $f(t)$ with Fourier Transform $F(s)$. So $$F(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f(t) \ dt$$ What is the Fourier Transform of $f'(t)$? Call it $G(s)$.So $$G(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f'(t) \…
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Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: $$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n}…
Mats Granvik
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Why do we use trig functions in Fourier transforms, and not other periodic functions?

Why, when we perform Fourier transforms/decompositions, do we use sine/cosine waves (or more generally complex exponentials) and not other periodic functions? I understand that they form a complete basis set of functions (although I don't understand…
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Real world application of Fourier series

What are some real world applications of Fourier series? Particularly the complex Fourier integrals?
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Fourier transform of function composition

Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually? I know you can do this for the sum, the product and the convolution of two functions. But I…
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How to interpret the adjoint?

Let $V \neq \{\mathbf{0}\}$ be a inner product space, and let $f:V \to V$ be a linear transformation on $V$. I understand the definition1 of the adjoint of $f$ (denoted by $f^*$), but I can't say I really grok this other linear transformation…
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Did Joseph Fourier ever make a pure mathematical mistake?

Cited by "Imre Lakatos and the Guises of Reason" John David Kadvany, 2001: It is remarkable that the nineteenth century was a time of error for mathematics: not trivial oversights or amateur conclusions, but fundamental mistakes in the…
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and some other situations where the Laplace transform…
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Non-power-of-2 FFT's?

If I have a program that can compute FFT's for sizes that are powers of 2, how can I use it to compute FFT's for other sizes? I have read that I can supposedly zero-pad the original points, but I'm lost as to how this gives me the correct answer. If…
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Derivative of convolution

Assume that $f(x),g(x)$ are positive and are in $L^1$. Moreover, they are differentiable and their derivative is integrable. Let $h(x)=f(x)*g(x)$, the convolution of $f$ and $g$. Does the derivative of $h(x)$ exist? If yes, how can we prove that $$…
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