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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103 views

Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
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1answer
53 views

Integral for $\frac{x}{x^2+1}cosx$

When computing Fourier transformation I came across these integral: $$ \int_{\Bbb R}\frac{x \cos x}{1+x^2}\;dx\text{ or } \int_{\Bbb R}\frac{x \sin x}{1+x^2}\;dx $$ Can anyone give me some hints on ...
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1answer
49 views

Is it always the case that lower frequencies contribute the most in a Fourier series?

Is it always the case that lower frequencies contribute the most in a Fourier series? Or to put it in other words, in the equation: $$f(t)=a_0+\sum^\infty_{m=1} a_m\cos \left(\frac{2\pi mt}{T}\right) ...
5
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1answer
90 views

Is there a complete orthornomal basis of a Hilbert space which takes positive values on a discrete set?

Is there a complete orthonormal basis $\{f_n\}$ (of continuous functions) of the Hilbert space of square integrable functions on $[0,\,\infty)$ for which there exists a countable set $S\subset ...
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2answers
54 views

zero distribution of the Fourier kernel $\Phi(u)$ for Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ The functional equation for $\zeta(s)$ is equivalent ...
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1answer
221 views

Taking the Fourier transform of a Hankel function

Considering the following inverse Fourier transform $$ f(t) = -\alpha \int_{-\infty}^{\infty} F(\omega)H_0^{(2)}(k(\omega) \beta) \exp(+j\omega t) d\omega$$ where $F$ is an arbitrary function and ...
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1answer
500 views

Is there a way to relate prime numbers and the fourier transform

According to what I know about Fourier transforms, any continuous periodic signal can be represented as a combination of sine and cosine functions. To me, this looks analogous to the "Fundamental ...
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1answer
35 views

inverse fourier transform of exponencial

Show that $F^{-1}(e^{-|x|}) =(\sqrt{2}/\sqrt{\pi})*1/(1+x^2)$ on $\mathbb R$. $F^{-1}$ is the inverse Fourier transform. Any help? how do you solve the integrals?
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1answer
80 views

Half-Fourier transform, relation to Delta function

so the Fourier transform of the Kronecker Delta function is (up to sign conventions / normalisation) $$\int_{-\infty}^\infty dt\; e^{i t \omega} = \delta(\omega).$$ Can one say anything about the ...
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1answer
39 views

Solution of the integral $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{3xy}{(x^{2}+y^{2}+z^{2})^{5/2})}e^{i(k_{x}x+k_{y}y)} dx dy$

I'm trying to solve the following integral: $$\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}\frac{3xy}{(x^{2}+y^{2}+z^{2})^{5/2})}e^{i(k_{x}x+k_{y}y)} dx dy$$ i'm using the solution ...
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0answers
44 views

complex integral with non integer power

I want to calculate this integral ...
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19 views

What is the best estimation for the following?

Let $f$ be 1-periodic and $f\in L_{p}[0,1]$ where $p>1.$ Let $D_{n}$ $n=0,1,2,..$ be the dyadic partition of $[0,1].$ Consider $$ F_{n}(x)=\frac{1}{|I^{n}_{j}|}\int_{I^{n}_{j}}f(t)dt, ...
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0answers
32 views

Obtaining generating function via Fourier transform

Series coefficient for a function can be obtained via Fourier transform: $$f^{(s)}(0)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} (- i \omega)^s \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt \, d\omega$$ ...
2
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0answers
542 views

Convolution of two Gaussians or two sinc functions using direct integration

I tried to solve the following to problems from Gaskil's book Linear Systems, Fourier Transforms, and Optics. But I'm struggling to get the right results. My experience with calculating convolutions ...
2
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0answers
48 views

Estimation of trignometric polynomial and Lipschitz estimation

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} ...
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0answers
38 views

how does the following parameters affect the peak ( radius ,theta) of the its frequency spectrum?

The following is the similar magnified picture to the second image that it indicates the parameter i am going to talk about. Suppose i am going to fourier transform (DFT) of the following picture ...
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3answers
142 views

Solution of Definite integral:$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}e^{i(k_{1}x+k_{2}y)}dxdy$

I'm trying to evaluate the following two dimensional integral: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}e^{i(k_{1}x+k_{2}y)}dxdy$ The paper that i'm ...
3
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1answer
107 views

Fourier transform of exponent?

Mathematica fails to find a Fourier transform of exponent. Yet according to this page $$\mathcal{F}[e^{2\pi iat}]=\delta(t-a)$$ and via substitution, $$\mathcal{F}[e^{at}]=\delta\left(t-\frac ...
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2answers
52 views

Two different results of Fourier Transform $xe^{-x}$

I have a function $f$ defined by $$f(x)=\begin{cases} xe^{-x} \textrm{ if } x>0,\\ 0,\textrm{otherwise}. \end{cases}$$ I wish to know the Fourier transform of $f$, i.e, $${\cal ...
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1answer
158 views

Can piecewise $C^{1}$ on $[a,b]$ imply Lipschitz continuity

I saw a statement that if $f$ is continuous,$2\pi$-periodic function which is $C^{1}$ piecewisely on $[-\pi,\pi]$, then its Fourier series converges uniformly to $f$ on $[-\pi,\pi]$. I was wondering ...
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0answers
136 views

Fourier transform of an exponentially singular radial function

I am trying to compute the 3D Fourier transform of a spherically symmetric function of the form $$f(r) = e^{\frac{1}{r} e^{-r}} - 1\, ,$$ which entails the integral $$\begin{aligned}F(k) =& \int ...
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70 views

Convergence of Fourier series in Sobolev space

So the problem is if $f\in H^{\frac{1}{2}}([0,1])\cap C([0,1])$, then $S_Nf$, the partial sum of fourier series converges uniformly to $f$. How would you show this by considering the quantity ...
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28 views

Two dimensional fourier transform computation

I have to compute the two dimensional fourier transform of the following function: $f(x,y,z) = \frac{x}{(x^{2}+y^{2}+z^{2})^{5/2}}$ the paper that i'm following defined the two dimensional fourier ...
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1answer
40 views

How to tell which function's Fourier series to use in order to calculate the value of series.

I got this question when I was doing some exercises. I was ask to establish $$ \sum_{n=0}^{\infty}\frac {1}{(2n+1)^2}=\frac{\pi^4}{96},\quad \sum_{n=0}^{\infty}\frac ...
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0answers
285 views

Convolution between Tempered distribution and schwartz function

$T$ is a tempered distribution on $\mathbb R$, $f$ is a Schwartz functions on $\mathbb R$. We define $T\ast f$ as $(T\ast f) (l)$=$T(f(-x)\ast l)$ for all $l$ Schwartz function, where the last $\ast$ ...
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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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1answer
86 views

On the convergence rate at infinity of the Fourier transform of the standard bump function

My question is concerned with the Fourier transform of the standard bump function, $\phi(x)=e^{-\frac{1}{x^2-1}}$ if $x\in (-1,1)$ and equal to $0$ if otherwise. As known, as $\phi$ has compactly ...
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0answers
37 views

Hopf Algebras Arising From Fourier Transforms?

At least for nice functions, a Fourier series is just a Laurent series on a circle (which means just substituting $z = re^{i\theta}$ into a Laurent series), so I can see Fourier analysis on Abelian ...
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2answers
151 views

How To Prove:$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4} = -\frac{7}{{720}}{\pi ^4}$

When I tried to solve this integral: $$\int_0^\infty {\frac{{{x^3}}}{{1 + {e^x}}}} \;{\rm{d}}x$$ I had trouble computing the sieries: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}$$ Thanks.
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1answer
127 views

Difference between $X(\omega)$ and $X(j\omega)$ notation of Fourier transform

In many reference materials i have come across fourier transform of a function $x(t)$ referred as $X(\omega)$ and $X(j\omega)$. But what is the difference between both the representations. Are they ...
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1answer
69 views

Spectrum of Rank 1 Operators

Given $\psi$ and $\phi$ in a Hilbert space $H$, we let $T$ be the rank-1 operator such that $$T\varphi=<\psi,\varphi>\phi.$$ It is easy to find the eigenvalues of $T$, they are $0$ and ...
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45 views

Momentum Representation vs Position Representation

I have a question involving the representation of operators in momentum representation and position representation. The question is a little long, so I'll do my best to explain it. We are given an ...
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1answer
122 views

Norm of Hardy-Littlewood maximal operator

We define Hardy-Littlewood maximal operator $M$ by \begin{equation} Mf(x)=\sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| dy \end{equation} where $B(x,r)$ denotes the ball centered at $x \in ...
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1answer
20 views

Fourier transform $\mathcal{F}_y\left\{\frac{e^{-|y|x}}{y}\right\}$

I have shown that $$\mathcal{F}_y\left\{\frac{e^{-|y|}}{y}\right\} = -i \sqrt{\dfrac{2}{\pi}} \tan ^{-1} (k)$$ where $k$ is the frequency variable. I need to find, however, ...
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44 views

Prove that $\prod_{k=0}^{n-1}cos\pi (x+k/n)=\frac{sin\pi n(x+1/2)}{2^{n-1}}$

$$ Prove\:that\: \prod_{k=0}^{n-1}cos\pi (x+k/n)=\frac{sin\pi n(x+1/2)}{2^{n-1}}\\given\:that\\\prod_{k=0}^{n-1}sin\pi (x+k/n)=\frac{sin\pi n(x)}{2^{n-1}}\\and\\sin(\phi+\pi/2)=cos(\phi) $$ From my ...
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1answer
55 views

Fourier parseval prove for misunderstanding of second negative exponent sign

See the picture below: I know if the sign is not '-', the following derivation can not continue,but I really want to know why $$e^{itx}\cdot e^{i\tau x}=e^{i(t-\tau)x}$$ How it can be that? I ...
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1answer
71 views

Is diagonal of sum matrix of rank one matrix and circulant matrix.

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in $\mathbb{R}^n, n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
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1answer
33 views

Computing integrals of inverse Fourier transform of function with compact support

Suppose $f \in C^{\infty}(\mathbb{R})$ with compact support, $f(0) = 1$ and derivatives satisfying $f^{(n)}(0) = 0$ for all $n = 1,2, \dots$. Consider \begin{align*} K(u) = \frac{1}{2 \pi} \int_{- ...
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0answers
48 views

Fourier inversion of an infinitely divisible multivariate gamma measure represented in polar form.

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...
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1answer
37 views

Expression for the Fourier transform of $f(x) = \frac{1}{1 +\|x\|^2}$ [duplicate]

I'm having troubles with the Fourier transform of $f(x) = \frac{1}{1 +\|x\|^2} \in L^2(\mathbb{R}^{n})$. For the case $n=1$ I got $\hat{f}(\xi) = \pi e^{-2\pi |\xi|}$ using residues. Does the general ...
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1answer
20 views

Showing $\cos{\pi \frac{mx}{R}}\sin{ \pi\frac{nx}{R}}$ are orthogonal in $L^{2}([0,R])$?

I'm trying to show that $\sin \left(\frac{n\pi}{R}x\right)$ and $\cos \left(\frac{m\pi}{R}x\right)$ are orthogonal using the trigonometric identity that $2\cos{mx}\sin{nx} = \sin((m+n)x) + ...
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1answer
696 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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2answers
183 views

Prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous

How does one prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous? I'm unsure of where to begin. Pertinent definitions: ...
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5answers
45 views

Proving a function $f(x + T)=k\;f(x)$ satisfies $f(x)=a^x g(x)$ for periodical $g$

I need to prove the following: If a function $\,f$ satisfies $$f(x+T)=k\;f(x), \forall x \in \mathbb R$$ for some $k \in \mathbb N$ and $T > 0$, prove that $\,f$ can be written as ...
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1answer
376 views

Fourier transform of squared exponential integral $\operatorname{Ei}^2(-|x|)$

Let $\operatorname{Ei}(x)$ denote the exponential integral: $$\operatorname{Ei}(x)=-\int_{-x}^\infty\frac{e^{-t}}tdt.$$ Now consider the function $\operatorname{Ei}(-|x|)$. ...
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1answer
96 views

Fourier Sine Series and Cosine Series

This is the Fourier Series representation for a periodic function with period 2p, given in my lecture note. $\dfrac{a_0}{2} + \sum_{n=1}^{\infty}(a_n cos(\dfrac{n\pi t}{p})+b_nsin(\dfrac{n\pi ...
0
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1answer
68 views

Prove the following relation between Fourier transform pairs

Lets say I have a function $f(x,y)$ and its corresponding Fourier transform $F(u,v)$. I'm trying to show that $f(x+by,dx + y) \leftrightarrow ...
0
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1answer
36 views

Transforming a function in the dirac delta function

I need help finding $k(\sigma)$ such that the family of functions $$ \delta_\sigma (x,y) = k(\sigma) e^{-\frac{1}{2}\frac{x^2 + y^2}{\sigma^2}} $$ defines the unit impulse $\delta(x,y)$ as $\sigma ...
2
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0answers
37 views

Is there a continuous and satisfying a certain condition but not integrable function?

Is there a continuous function $f$ on the interval $[0,1]$ which satisfies $$ |f(x+h)+f(x-h)-2f(x)|\leq \mathrm{const} \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta ...
0
votes
1answer
22 views

Seperable functions have seperable Fourier transforms

How do I show that if $f(x,y)$ is separable into a product of a function of $x$ and a function of $y$, its Fourier transform $F(u,v)$ is also separable into a function of $u$ and a function of $v$?