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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Essential self-adjointness of the Laplace operator via the Fourier transform

I'm working through some notes on showing the essential self-adjointness of the Laplace operator on $\Bbb R$ via the Fourier transform (see here) but there seems to be a little bit of liberty taken at ...
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How do you verify this identity of fourier transform involving $\delta(x-x')$

How do you prove that $$\delta(x-x') = \frac{1}{2\pi} \int\limits_{-\infty}^{\infty} e^{-ik_x(x-x')} \, \mathrm{d} k_x$$ Attempt, take fourier transform of delt function and using the sifting ...
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Variation of Delta function integrated

We all know: $$\int_0^\infty \delta(y) dy = 1.$$ How about $$\int_{-\infty}^\infty y\delta(y) dy .$$ The solution of this is $0$. I have no idea how to get this. thx,
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Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
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Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
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Converse to the Hausdorff-Young inequality

Let $p^{\prime}, q^{\prime}$, and $s^{\prime}$ denote the conjugate exponents of $p,q,$ and $s$ respectively, i.e., $\frac{1}{q^{\prime}} + \frac{1}{q} = 1$. The Hausdorff-Young inequality says that ...
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Relation between Hankel transform and Fourier transform

As a physics student, I ran into the following problem. I left out a lot of context, if anything is unclear please ask me. I quote: The statistic that is observable is the angular correlation ...
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How to evaluate this (Fourier) integral? [duplicate]

Does somebody know how to evaluate $$\int_{\mathbb{R}^n}\frac{e^{i\langle\xi,x\rangle}}{\|\xi\|_2^2}d\xi$$ for some given $x\in\mathbb{R}^n$ and $n\in\{1,2,3\}$?
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Prove the following vectors are linearly independent

So I have these three vectors: [i, 2+i, 3]; [2, -i, 4-i}; [3, -1, 2] and I need to show they are linearly independent. This means that given scalars $x_1, x_2, x_3$ their scalar sum should equal 0. ...
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Given $\ f(\theta)=\theta(\pi-\theta)$ is a $2\pi$-periodic odd function on $[0,\pi]$. Compute the Fourier coefficients of $f$, and show that $\ f(\theta)=\frac{8}{\pi} \sum_{\text{$k$odd} \ \geq 1} \... 1answer 42 views Find Fourier Coefficients I am asked to find the coefficients for$f(t)=\sin^{2}(5t)$$$Period =\frac{\pi}{5}$$ so I wrote $$a_n\cdot\sin(\frac{n\pi{t}}{\frac{\pi}{10}})=\sin^{2}(5t)$$ $$a_n\cdot\sin(10n{t})=\sin^{2}(5t)$$ ... 0answers 82 views Saturation of the Babenko–Beckner inequality The Babenko-Beckner inequality$|| \mathcal F f ||_q \geq C(q,p)||f||_p$is a well-established theorem. It relates the$q$-Norm of a Fourier transform$\mathcal F f$of a function$f$to its$p$-... 1answer 46 views Confused about Fourier series? From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ... 2answers 59 views Does these sequence and series converge? Let$f\in C^1[-\pi,\pi]$st$f(-\pi)=f(\pi)$and define $$a_n=\int^{\pi}_{-\pi} f(t)\cos nt dt\,$$ for$n \in\Bbb{N}$. Then does the sequence$\{na_n\}$converges? And does the series$\sum^{\infty}_{...
Take for example one period of a sine: $f(x) = \{\sin(\omega_1x) \; \mathrm{if} \; x \in [0, 2\pi) \;; \quad 0 \; \mathrm{elsewhere} \}$ If we now translate $f$, then according to the argument that ...