# 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).

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### Why do we call a Fourier series a series of complex exponentials?

Euler's formula is $$e^{{ix}}=\cos x+i\sin x$$ I have recently learned that a periodic function can be decomposed into an infinite series of cosines and sines. However, in the formulas that I've seen,...
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### Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
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### Dirac Delta or Dirac delta function?

Is Dirac delta a function? What is its contribution to analysis? What I know about it: It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
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### Criteria for swapping integration and summation order

I have a function (a potential from an electrostatic potential via a Fourier series) in the form of $$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m) du\, dv$$ ...
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### Discrete Fourier Transform: Effects of zero-padding compared to time-domain interpolation

While studying the various algorithm implementations available on-line of the Fast Fourier Transform algorithm, I've come to a question related to the way the DFT works in theory. Suppose you have a ...
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### Is deconvolution simply division in frequency domain?

Is it correct to say that deconvolution simply division in frequency domain? And that convolution in time domain is multiplication in frequency domain. And is it a convention to notate a function in ...
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### Derive Fourier transform of sinc function

We know that the Fourier transform of the sinc function is the rectangular function (or top hat). However, I'm at a loss as to how to prove it. Most textbooks and online sources start with the ...
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### What is a good book, or article, that explains the history of fourier analysis?

What is a good book on the history of Fourier Analysis? I'm looking for a book which explains how it came to be and what the mathematicians (or physicists) were thinking when they came up with it. If ...
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### Relation between function discontinuities and Fourier transform at infinity

I have made the following assertion a few times in this space without ever having provided a proof: Let $m$ be the smallest number such that a function $f \in L^2(\mathbb{R})$ has a discontinuity in ...
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### Where is the wild use of the Dirac delta function in physics justfied?

Wikipedia has a wild article about the Dirac delta function. Are the things listed correct? Or is there no proof that they are correct? For my master thesis I want to refer to rigorous proofs of these ...
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### A bound on the Fourier coefficients of an $\alpha$-Lipschitz function

I am asked to show that if $0 < \alpha < 1$, and if $f \in \Lambda^\alpha(\mathbb{T})$, then we have for $k\neq 0$, $$|\widehat{f}(k)| \leq \pi^\alpha \frac{\|f\|_{\Lambda^1}}{k^\alpha}$$ I ...
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### What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...
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### DFT - Why are the definitions for inverse and forward commonly switched?

Sometimes the forward DFT is defined with a negative sign in the exponent, sometimes with a positive one and occasionally with a $1/N$ coefficient. I see this all over the place online. I don’t see ...
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### Simple proof that Fourier transform is an isomorphism between $L^p$ spaces for $p \neq 2$?
It is known that the Fourier transform $\mathcal F$ maps $L^2 \to L^2$ as an (isometric) isomorphism and $L^1 \to L^\infty$ as bounded operator. Via Riesz-Thorin this result can be extended to give ...