# 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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### Given Fourier coefficients of a function , find the function

Given these Fourier coefficients: $$X[k]=\begin{cases} 1 & \text{, k even}\\ 2 & \text{, k odd}\\ \end{cases}$$ I want to find the analytical expression for the function. What i tried was ...
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### Dilation of Fourier transform

Let $f\in \mathcal{S}(\mathbb{R}).$ The Fourier transform of $f$ is defined by $\hat{f}(w) := \int_{-\infty}^\infty f(x) e^{-2\pi i x w} dx$. We use the notation $f(x) \longrightarrow \hat{f}(w)$ to ...
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### Fourier transform of Si[$x^2 + y^2$]; Energy integrals involving sin integral functions

Problem Statement I'm trying to prove( or disprove ) the following identity \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\Big[\text{Si}[x_1^2 + y^2]- \text{Si}[x_2^2 + y^...
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### Class of Functions , that admit Fourier transforms

For which class of functions/distributions is it sensible to take a Fourier transform ?
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### Several questions about the proof of this lemma

I am reading this paper. I have Several questions about the proof of the lemma 3.3. How to Prove 1,2,3,4?
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### Integration and differentiation of Fourier series

I am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose that ...
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### Fourier Series and Fourier Transform confusion.

I dont understand the following paragraph after the proof. In particular, how does that theorem above give us that the Fourier transform maps $L^2$ onto $l^2$? all that theorem says is that this set ...
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### Deconvolution by disks

I have a function $$f(x,y) = \begin{cases} 1, \|(x,y)-p\| < r\\0, \|(x,y)-p\|\geq r\end{cases}$$ where $p$ is some unknown point in $[0,1]^2$; i.e. $f$ is the characteristic function of some disk ...
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### Fourier sine and cosine series: reconstruction is shifted with respect to measured data

I am working in strain analysis. Strain in a mechanical testing machine is captured by strain gages. Signals are like the slim line in the graph below showing strain versus time. The data are of the ...
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Let $u:\mathbb R \times \mathbb R \to \mathbb C$ be some function so the everything in the following make sense. Consider the following PDE: $\frac{\partial^{2}}{\partial t^2} u(x,t) + \frac{\... 0answers 27 views ### A$H^p$function Set$\mathbb U=\{x+iy|\;y>0\}$. A function$f:\mathbb U\to\mathbb C$is called a$H^p$function if$f(z)$is holomorphic and$\|f\|_{H^p}:=\sup_{y>0} \left(\int_{-\infty}^{\infty} |f(x+iy)|^p dx\...
The Laplace transform of a function $f(t)$ is the projection of $f(t)$ vector (indexed with $t$) onto the linearly independent set of vectors $e^{st}$. The projection of a vector $\vec{v}$ onto ...