# Tagged Questions

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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### How to prove that inverse Fourier transform of “1” is delta funstion?

$\mathscr{F}\{\delta(t)\}=1$, so this means inverse fourier transform of 1 is dirac delta function so I tried to prove it by solving the integral but I got something which doesn't converge.
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### How do you prove that $\lim f(x) = 0$, when $f$ is rapidly decreasing?

Let $f: \Bbb{R} \to \Bbb{R}$ be rapidly decreasing in the sense than $\sup_{x \in \Bbb{R}} |x|^k |f^{(\ell)}(x)| \lt \infty$ for all $k, \ell \geq 0$, where $f^{(\ell)}$ is the $\ell$th derivative. ...
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### Magnitude of Fourier coefficients when $||f||_2 \leq 1$

Let $f \in L_2[- \pi, \pi]$ so that $||f||_2 \leq 1$. Can I say anything about $f$'s Fourier coefficients' magnitude without assuming anything else about $f$? To be more accurate: The squared norm ...
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### Is there any handwavy argument that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$?

It should not be a good argument but rather a short one and one that convinces a physicist ( so no need for mathematical rigor ) that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$ ...
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### Representing real function as integral over trigonometric functions

Since one can clearly express any function $g(x)$ as $$\int_0^{\infty} A(k)\cos(kx)dk+\int_0^{\infty} B(k)\sin(kx)dk,$$ how would $G(k)$ relate to $A(k)$ and $B(k)$? In other words, what would how ...
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### How to perform division in modular arithmetic on complex exponentials: controversy, bugfix required

I have a complex exponent with prime divisor 7: $e^{\frac{2\pi i \cdot 2}{7}}$ and want to take it to the power 1/3: $e^{\frac{2\pi i \cdot 2}{7 \cdot 3}}$ (I'm learning, how to work with division ...
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### Fourier transform of a complex exponential with quadratic argument

I'm a PhD student who is starting to work right now in the well-established field of ultra-fast optics. The thing is that, in most of the papers I have been reading during the past few days, there is ...
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### Compare mixed derivatives to laplacian

Suppose $u,f$ periodic and smooth in $Q=[0,1]^n$ such that $\Delta u=f$. Show that for each $i,j$, $$\int_Q \left| \frac{\partial^2 u}{\partial x_i \, \partial x_j} \right|^2 \leq C \int_Q |f|^2.$$ ...
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### What are the properties of the fourier transform of a phase-only function?

Given a function of the form: $$f(x) = e^{i\phi(x)} | \phi(x)\in\Re$$ What are the properties of its Fourier transform? For instance, purely real functions have Fourier transforms with symmetric ...
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### Constructing an L2 function from an entire function bounded on R

I have an entire function $f(z)$ of exponential type $\tau\geq0$ that is bounded on $\mathbb{R}$ and zero at every member of the complex sequence $\{\lambda_n\}$. What I want is an entire function of ...
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### Q: Calculating Fourier Coefficients and Inverse Fourier Transform

Let $\Omega >0$ and $x \in \mathcal{B}_{\Omega/2}$ is continuous. Define $\hat{y}(\omega) = \sum_{n \in \Bbb Z} \hat{x}(\omega - n\Omega)$. If $\hat{y}$ is expressed as \hat{y}...
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