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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41 views

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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1answer
81 views

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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127 views

Example of a not good kernel

Background Given a sequence ${K_n}$ of functions on region $T$. Then ${K_n}$ is a good kernel if: (1) $\frac{1}{2\pi}\int_{-\pi}^{\pi} K_{n}(x)dx = 1$ $\forall n \in N$ (2) $\frac{1}{2\pi}\int_{-\...
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1answer
39 views

Is $C_c$ dense in $L_p$ for $0<p<1$?

Let $C_c$ be the set of compactly supported functions on $\mathbb{R}$ that are infinitely differentiable. Let $S$ be the set of Schwartz functions. It is well known that $C_c$ (hence $S$) is dense in $...
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201 views

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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1answer
134 views

Fourier transformation on a torus and the definition of fractional Laplacian

as we know, in $R^n$, for a function $f$, we can define its Fourier transform as $$\hat f(\xi)=\int_{R^3}f(x)e^{-ix\cdot \xi}d x,$$ with this, the Laplacian of $f$ can be elegently defined by $$\...
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88 views

Taking the Fourier Transform of a specific function.

I have this function: $$f(x) = \prod_{p\text{ is prime}} \left(1 - \frac{x^2}{p^2}\right)$$ Now, this function can be said to be an infinite degree polynomial with zeros on each of the primes and ...
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39 views

Is it feasible to think of laplace transform and z transform as projections?

For Fourier transform, it has been ingrained in my head that all we are doing is projecting a function onto its Fourier basis, namely $(1, cos(t), sin(t),...cos(nt), sin(nt) ...)$ Can anyone comment ...
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118 views

Proving orthogonality of complex form of Fourier Series

I am lost when working on this complex Fourier Series question, I am sure it is a basic simple problem but I am not well versed in applied math: Show that $\{e^{\mathscr i n\pi x/\mathscr l}\}, n =...
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40 views

Help figuring out output signal of LTI system.

Would greatly appreciate any help in figuring out the output signal of my discrete time LTI system. My input signal is cos(ωn) and my frequency response is H(e^jω)=(1+e^−jω)/2.
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96 views

Convolution Properties

I have a quick question about certain algebraic properties of convolution. If I have 3 functions $f(x)$, $g(x)$ and $h(x)$, is the following true? $\Big[ f(x) . g(x)\Big] \circ h(x) = \Big[f(x) \circ ...
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1answer
51 views

Fourier cosine and sine transform of $\exp{(-ax)}(1+bx)^{-1}$ and $\exp{(-ax)}(1+bx)^{-2}$

As stated in the title I should calculate the cosine and sine Fourier transform of: $$f_1(x)=\exp{(-ax)}(1+bx)^{-1}$$ and $$f_2(x)=\exp{(-ax)}(1+bx)^{-2}$$ That obviously means calculating: $$\...
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1answer
55 views

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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1answer
113 views

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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1answer
504 views

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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31 views

Dual formulation of weak $L^p$

Let $1<p\leq \infty$. Then we have$$||f||_{L^{p, \infty}(X,d\mu)} \sim_psup\{\mu(E)^\frac{-1}{p'}|\int_E f d\mu|:0<\mu(E)<\infty\}$$ Where$||f||_{L^{p, \infty}(X,d\mu)} = \sup_{\lambda\in\...
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1answer
87 views

Frequency response of a linear, shift-variant system

I am working my way through recorded lectures and a textbook related to DSP, and have come across a question that I am not sure how to answer. This is probably just due to how new I am to these ...
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1answer
86 views

Fourier series of rescaled cosine function

How would I find the Fourier series of $\cos\left(\, 5x/2\,\right) $ on $\left[-\pi,\pi\right]$? Progress $$A_0={1\over 2\pi}\int_{-\pi}^\pi \cos(5x/2)dx={2\over 5\pi}$$ $$A_n = {1\over \pi} \int_{-...
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1answer
57 views

Why does the integral equal $1$?

Let $a\in\mathbb{R}-\mathbb{Z}$. Why is the following equality true? $$1 = \frac{1}{2\pi} \int_0^{2\pi} \left| e^{-i(\pi-x)a} \right|^2 dx$$ More precisely, why is the integrand equals $1$?
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62 views

Absolute square in deriving Fourier transform variance

I'm having some trouble understanding how to derive the variance of the Fourier transform. This is for an image, i.e., it's a 2D transform. The variance is $|\hat{I}(\xi,\eta)|^2$, the absolute ...
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1answer
159 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega t}e^{\...
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2answers
75 views

If $f \in L^2(\mathbb T)$ then $S_n(f) \to f$ in $L^2$ sense.

Theorem: If $f \in L^2(\mathbb T)$, then $S_n(f) \to f$ in $L^2(\mathbb T)$ sense. Proof: Let $f \in L^2(\mathbb T)$, then by definition $\|f\|_2^2 = \frac{1}{2\pi} \int_0^{2\pi} \vert f(x) \vert^2 \,...
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0answers
559 views

Generalization of the Jacobi-Anger expansion to higher harmonics

I know the Jacobi-Anger expansion relation which gives the Fourier development of $e^{i z \cos(\theta)}$ and ${ e^{i z \sin(\theta)} }$, such that $$ \begin{cases} e^{i z \cos(\theta)} = \sum\limits_{...
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2answers
183 views

Analog of the Fourier transform on a bounded domain?

Suppose $\Omega \subset \mathbb{R}^d$ is a bounded, simply-connected domain with whatever other "niceness" properties are necessary for this question to make sense. Define the "Fourier transform" over ...
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1answer
29 views

If $u : \Bbb R \to \Bbb R$ satisfies $u' + 2\pi x u = 0$, why does $\hat{u}$ (the Fourier transform) also satisfy this?

I'm trying to understand why if a function $u : \Bbb R \to \Bbb R$ satisfies the differential equation $u' + 2\pi x u = 0$, then so does the Fourier transform. The properties I have that I can use ...
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61 views

Fourier transform of $e^{-\frac{x^2}{2}}\frac{1}{\mathsf{sinc}( x )} $

I have to find inverse Fourier transform of \begin{align*} F(\omega)=e^{-\frac{\omega^2}{2}}\frac{1}{\mathsf{sinc}( \omega )} \end{align*} I was wondering whether it exists maybe in a sense of ...
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1answer
249 views

Question about a proof of Riemann localization theorem

The Riemann Localization Theorem states that Let $f \in L_{2 \pi}^2$ and $x_0 \in \mathbb R$. Then $$ \lim_{n \to \infty} (S_nf)(x_0) = f(x_0)$$ if and only if there is a $\delta \in (0, \pi)$ ...
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39 views

Sum of unitary complex numbers

Let us define: $$\varphi(x,n,t):=\frac{1}{n}\sum_{y=1}^n \left| \sum_{k=1}^n e^{2ik\pi (x-y)/n + 2i \sin(2k\pi/n) t} \right|$$ Does somebody have an idea how to prove that $$ \sup_{x=1,...,n} \...
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43 views

integration and convolution

Please can some one help me on the following integration. $$ G(\nu)=\frac{1}{\Delta t}\int_{t_a - \frac{\Delta t}{2}}^{t_a + \frac{\Delta t}{2}} f(t_a -t)e^{-2\pi\nu it}dt $$ where $f(x)=\mbox{...
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1answer
64 views

Does there exist a non-negative valued compactly supported function such that its Fourier transform only vanishes at a given point?

My question is as follows: Given $t_0\in\mathbb{R}$. Does there exist a non-negative valued compactly supported function $f\in L^1(\mathbb{R})$ such that its Fourier transform, $\widehat f\left( t \...
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29 views

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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36 views

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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200 views

Fourier series of a periodic odd function

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} \...
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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 $$a_n\cdot\sin(\frac{n\pi{t}}{\frac{\pi}{10}})=\sin^{2}(5t)$$ $$a_n\cdot\sin(10n{t})=\sin^{2}(5t)$$ ...
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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 $p$-...
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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 ...
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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}_{...
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1answer
66 views

Fourier transform: no time localization but an inverse exists. How can these properties go together?

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 ...
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42 views

Find Coefficients from already fourier function

Hello I have this function and I'm asked 1.Find the period for $f(t)$ 2.Find the coefficients $a_n$ and $b_n$ $$f(t)=2(cos(2t+\frac{\pi}{4})-sin(6t-\frac{\pi}{2}))$$ I know that the period for $...
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2answers
99 views

Fourier Transform of $f(x) = \exp(-\pi ax^{2} + 2\pi ibx)$

I was trying to take the FT of $$f(x) = \exp(-\pi ax^{2} + 2\pi ibx)$$ This is just the shifting rule applied to the FT of $$g(x) = \exp(-\pi ax^{2})$$ which is given by $$\hat g(k) = \frac{1}{\...
3
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1answer
331 views

What is the relation between analytical Fourier transform and DFT?

First of all let me state that I searched for this topic before asking. My question is as follows we have the Analytical Fourier Transform represented with an ...
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1answer
53 views

How is the exponential in the Fourier transform pulled out of the integrand?

I'm looking at Fourier Transforms in a Quantum Physics sense, and it's useful to associate the Fourier Series with the Dirac Delta. The book I'm using follows this argument (Shankar, Quantum Mechanics)...
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32 views

Harmonics conditions for a plucked string

Given a plucked string which is taken on the interval $[0,\pi]$, and it satisfies the wave equation with $c=1$. The initial position of the string is: $\ f(x) = \frac{xh}{p}$ ($0\leq x\leq p$), and $\ ...
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1answer
43 views

Showing $\sum_{n=-\infty}^{\infty}\exp\left(-\pi an^2+2\pi ibn\right)=a^{-\frac{1}{2}}\sum_{m=-\infty}^{\infty}\exp\left(-\frac{\pi(m-b)^2}{a}\right)$

How do I show that \begin{align} \sum_{n=-\infty}^{\infty} \exp\left(-\pi a n^2 + 2 \pi i bn\right) = a^{-\frac{1}{2}} \sum_{m=-\infty}^{\infty} \exp\left(-\frac{\pi(m-b)^2}{a}\right) \end{align} is ...
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2k views

Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
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104 views

$f\in L^1\cap L^2$ implies $\hat f \in L^1$?

Given $f\in L^1(\mathbb R^d)\cap L^2(\mathbb R^d)$. The Riemann-Lebesgue lemma and the unitarity of the Fourier transform on $L^2$ implies that $\hat f \in L^2\cap C_0$ where $C_0$ are continuous ...
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120 views

strange transform of dirac delta function

one of our homework solutions states that $$\delta(x)\equiv\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega e^{-i\omega x} $$ is the Fourier transform of the dirac delta function. But according to the ...
2
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1answer
86 views

On the proof of Fejér-Riesz theorem

I'm having a course about Analytic Number Theory, and I'm having trouble understanding the proof of Fejér-Riesz Theorem: http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf First of all, I didn't ...
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1answer
53 views

Inverse Fourier transform of $F(\omega)=\frac{1}{\sum_{k=1}^N e^{i \omega z_k}}$

I am looking for the inverse Fourier transform of \begin{align*} F(\omega)=\frac{1}{\sum_{k=1}^N e^{i \omega z_k}} \end{align*} where $z_k \in Z$. But I don't know how to approach it. This reminds ...