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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Find coefficients of polynomial that has zeros at certain points

Given a list of values z0, z1, ..., zn-1 (possibly with repetitions), show how to find the coefficients of a polynomial P(x) of degree-bound n + 1 that has zeros only at z0, z1, ..., zn-1 (possibly ...
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How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0)$$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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Let $F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha$. Is it true $F, F^{-1}\in L^{1}(\mathbb R)$?

Define $$F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha, \ (x\in \mathbb R).$$ It is clear to me that, the integral converges for every real $x$ (as near origin integrand is ...
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Spatio-temporal triple correlation

I would like to simplify if possible the spatio-temporal triple correlation of the following function: $$f(\vec{x},t)=\delta(\vec{x}-\vec{x}_0(t)) \otimes f_p(\vec{x})$$ where $\delta$ is the Dirac ...
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Inner product of functions is preserved by inner product of fourier co-efficients, more to plancherel theorem

By Plancherel theorem the map $T:L^{2}(\mathbf{T}) \longrightarrow l^2(\mathbf{Z})$ defined by $T(f) =(f^{\wedge}(n))_{n \in \mathbf{z}}$ is a surjective isometry. But I have to show a bit more.That ...
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Why are almost all prime sized circulant matrices non-singular?

Consider an $n$ by $n$ circulant matrix whose values are either $1$ or $0$. Now let $n$ be a prime. For any such $n$ there are exactly two circulant matrices that are singular (over $\mathbb{R}$). The ...
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Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
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How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
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Integral of a function is not affected by altering the function values at zero-measure set

I'm studying about Fourier analysis from a book Fourier analysis and its applications, Folland 1992 and I have one point in the source I need clarification about: On page 69 it is stated that: "The ...
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A quantity depending on two independent variables must be a constant, why?

I'm studying about Fourier analysis and there is one part in my book about partial differential equations I don't understand. It states that a quantity, which depends on two independent variables $x$ ...
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find the sum $\sum_{k=1}^{\infty} \frac {\sin^2(kx)}{k^2}$, for $x\in [-\pi,\pi]$

find the sum $$\sum_{k=1}^{\infty} \frac {\sin^2(kx)}{k^2},$$ for $x\in [-\pi,\pi]$. Also i know that $$\sum_{k=1}^{\infty} \frac {\sin(kx)}{k}=\frac {\pi-x}{2}$$ Any help would be much appreciated.
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Problem with the Fourier transform of a function

I'm having some troubles with this one: $$\mathcal F(e^{-|x|} +|x|e^{-|x|})$$ I know that $\displaystyle\mathcal F(e^{-|x|})={1\over \pi (1+w^2)}$ but the second part is where I get stuck.
I'm trying to convert this phasor into a sinusoidal waveform. $$j6e^{-j\pi/4}$$ Here's what I have so far: $$6\sin(\omega t-\pi/4) = 6\cos(\omega t-\pi/4 - \pi/2) = 6\cos(\omega t-3\pi/4)$$ ...