# 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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### Shifted Fourier transform

Please can some one help and give me a direction to evaluate the following shifted Fourier transform: \begin{alignat}{2} s(x_c) =&\frac{1}{\Delta x_0} \int_{x_c-\Delta x_0}^{x_c+\Delta x_0}...
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### Fourier Differentiation Property

I have been given this problem to solve: Define the function f(t) by $$f(t) =\begin{cases} e^{-kt},& t \geq 0 \\ 0,& \text{otherwise}\end{cases}$$ where $k > 0$ is a real number. ...
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### Convergence of Fourier Sine Series for Gerneral Continuous Function

This is my question: How do I should that, for $f \in C[0,\pi]$ with $f(0) = f(\pi) = 0$, the Fourier sine series $$\tilde f_n = \sum_{r=0}^n b_r \sin(r s)$$ converges uniformly to $f$ on $[0,\pi]$...
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### Some issues for solving differential equations using Fourier transform

Fourier transform is a powerful tool for solving differential equations. But I don't really know when the Fourier transform will give us the full general solution if it can be used. A simple example ...
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### Fourier Transform of $f(t) = e^{-kt}$

I am trying to calculate the fourier transform of the following function: $$f(t) =\begin{cases} e^{-kt},& t \geq 0 \\ 0,& \text{otherwise}\end{cases}$$ where $k > 0$ is a real number. ...
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### Understand this Fourier transform $\int \frac{1}{|x|}e^{ikx} d^3 x = \frac{4 \pi}{k^2}$

I found the equation $$\int \frac{1}{|x|}e^{ikx} d^3 x = \frac{4 \pi}{k^2}$$ in a 'physics' textbook and I just don't understand what this equation tries to tell me. Is there anybody who ...
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### Finding multiple functions with same $f_{even}$ but different $f_{odd}$?

A function can be decomposed as $f(x) = f_{even}(x) + f_{odd}(x)$ where $f_{even}(x)=\dfrac{f(x)+f(-x)}{2}$ and $f_{odd}(x)=\dfrac{f(x)-f(-x)}{2}$. If we know only $f_{even}$, how can we find ...
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### Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

Let $A$ be a bounded or unbounded selfadjoint linear operator on a complex Hilbert space $H$ with spectral representation $A=\int_{\sigma}\lambda \, dE(\lambda)$ given by the Spectral Theorem for ...
Exercise 5: If $f\in L^1$ and $\int |t\hat{f}(t)|<\infty$, prove that $f$ coincide a.e. with a differentiate function whose derivative is $i\int_{-\infty}^{\infty}t\hat{f}(t)e^{ixt}dt$ I know a ...