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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334 views

Complex-valued Fourier integral: $ \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $

I'm working on the Fourier transform, but I don't know how to evaluate the integral: $$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$
2
votes
1answer
109 views

$\int_{-\infty}^{\infty}\!e^{- \pi (x+iy)^2}\,dx = 1$ for all $y$.

Can anyone provide a proof of why $\int _{-\infty} ^ {\infty} e^{-\pi (x+iy)^2} dx$ equals 1, for all y ? $x$ and $y$ are real numbers. EDIT: We already know this for y=0. Thank you.
2
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1answer
104 views

Need help with Fourier transform problem

I'm trying to calculate the Fourier transform of the unit step function, $$\mathcal{F}[u(t)] \ = \int_{-\infty}^{\infty}u(t)e^{-i\omega t}dt \ = \int_{0}^{\infty}e^{-i\omega t} dt. \tag{1}$$ This ...
1
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2answers
47 views

Find triple functions $ (g_0,g_1,g_2)$ such that $g_0+g_1'+g_2'' = \delta_0-\delta_1$

I want to find a triple of compactly supported continuous functions $ (g_0,g_1,g_2)$ on $\mathbb{R}$ such that $$g_0+g_1'+g_2'' = \delta_0-\delta_1$$ This is seemingly not so hard but ive broken my ...
1
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1answer
177 views

Parseval's identity

How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
2
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1answer
79 views

Fourier analysis exercise

I need a hand with this question: If $f\in{L_1(\mathbb{R})}$ and $g\in{L_2(\mathbb{R})}$, then prove that $\widehat{f*g}=\hat{f}\cdot \hat{g}$ As a tip, i have been told to prove that: ...
0
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1answer
186 views

Fourier transform of integral operator

I would like to know which is the fourier transform of integral operator: $$Tf(x)=\int_{-\infty}^{+\infty}\quad f(x)dx\rightarrow \hat{T}\hat{f}(k)$$ I know that (is it right?): ...
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1answer
55 views

Fourier Analysis of a Time Series

The picture below displays experimental data on concentration oscillations in a chemical reaction. I would like to find frequency characteristics of the series. More specifically, assuming that the ...
4
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1answer
199 views

Mathematically inclined books on Signal Processing Theory

First off, i know this may seem off topic but i could not find help in signal processing communities so i was hoping there would be people here who both love mathematics and have interest in signal ...
0
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1answer
43 views

Equivalent of definite integral in fourier space

I would like to know the equivalent in Fourier space of an double integral over a circular domain: $$\int\int_C f(x,y) =\int_0^l \rho d\rho\int_0^{2\pi}d\theta f(\rho,\theta)\rightarrow ????????$$ I ...
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0answers
71 views

Inverse fourier transform involving exponentials

I would like to calculate the inverse fourier transform of: $$\hat{f}(k)=\exp(-a k^2+ikv)\cdot \frac{\sinh(m\sqrt{(b+ck^2+ikf)^2-d})}{\sqrt{(b+ck^2+ikf)^2-d}}$$ Any clue? Thanks!
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0answers
117 views

Discrete fractional fourier transform

I have written a code for producing matrix of fractional fourier transform with the help of eigen vectors of fourier transfom matrix. Does anyone know the elements of this matrix ( for example a 4 by ...
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1answer
116 views

How to integrate $f(x)$?

I've been asked to integrate $\int{f(x)}^2 dx$ between the ranges of $L$ and $-L$. I'm stuck! I understand how to integrate a constant or a function as in $x^2$ or something, but the $f(x)$ format is ...
0
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0answers
77 views

Intregral of exponential of Shannon Entropy Function

Here I am going to ask a similar question as rde asked , that is what is the integral of exponential of entropy function. That is what is the value of $F[H(x)]=\int_{-1}^{+1} e^{ikH(f(x^2))} dx$ ...
2
votes
1answer
200 views

Verifying Convolution Identities (Not yet Resolved! Help please!)

Note: I don't yet have a solution to my main issue yet which I have elaborated on in the edit. Further attention is deeply appreciated. :> $\bf{\text{Original Question}}$: Let $G$ be a locally ...
0
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1answer
105 views

A basic question about $\operatorname{supp}f$ (support of f).

Is it true that $\operatorname{supp}f$ is the complement of the biggest open set where $f=0 $? Here $\operatorname{supp}f=$ {$x\in \Bbb R^n ; f(x)\not=0$} and $f\in C$ (collection of continuous maps ...
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0answers
93 views

P.d.f of a discrete fourier transform of binary variables

Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$. The discrete fourier transform is defined $b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
5
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2answers
1k views

Numerical Approximation of the Continuous Fourier Transform

Given a function $F(k)$ in frequency space (sufficiently nice enough, eg. a Gaussian), I would like to compute its Fourier inverse \begin{equation}f(x) = ...
2
votes
1answer
160 views

In my Fourier text book, there are the following exercises to prove. why do some of them have the same left side but have different right sides?

In my Fourier text book, there are the following exercises to prove.why do some of them have the same left side but have different right sides? The demand of these question is to prove these ...
3
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1answer
105 views

Dense set in $L^2$

Let $ \Omega\subset \mathbb{R}^n$ with $m(\Omega^c)=0 $. Then how can we show that $ \mathcal{F}(C_{0}^{\infty}(\Omega))$ (here $ \mathcal{F}$ denotes the fourier transform) is dense in $L^2$(or ...
21
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4answers
664 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
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1answer
61 views

Estimate derivatives in terms of derivatives of the Fourier transform.

Let us suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a smooth function. Furthermore, for every $\alpha$ multi-index, there exists $C_\alpha > 0$ such that $$ |D^\alpha f(\xi)| \leq ...
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1answer
190 views

Fourier transform of $\frac{1}{1+x^2}$

we know that the Fourier transform of $\frac{1}{1+x^2}$ is $f(y) = \int_{-\infty}^{\infty} \frac{1}{1+x^2} e^{-2\pi i x y} dx $ . Here is the idea used in my textbook, for y<0 : We calculate the ...
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0answers
37 views

About this question,I have done something,but I can't go on.

About this question,I have done something,but I can't go on.Firstly,you can examine the validity of my work.If it is right,please go on to compute.If it is a wrong way,please give me your idea.
2
votes
1answer
56 views

Proving that a certain function is in $W^{1,n}(B(0,1))$

Fix $0<\alpha<1-\frac{1}{n}$ and let $f\colon\mathbb{R}^n \rightarrow \mathbb{R}$ be the function $f(x)=(\log(\frac{1}{|x|}))^{\alpha}$. How can I prove that $f\in W^{1,n}(B(0,1))$?
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0answers
54 views

To construct a Schrödinger wave with prescribed mean position and momentum

Let $\lambda_x, \lambda_\xi>0$ be fixed parameters which we will call position lengthscale and momentum lengthscale respectively. Fix a (position-)point $x_0\in \mathbb{R}$ and a (momentum-)point ...
1
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1answer
35 views

Confusion of one definition in Fourier analysis

The symbol occurs on Page 22 of Bahouri's book Fourier analysis and nonlinear differential equations. As defined there, $$f(D)a:=\mathcal{F}^{-1}\{f\mathcal{F}a\}.$$ The question comes from the ...
2
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2answers
110 views

Show existence of a continuous $k$ on $\mathbb{R}^2$ such that $(u,\phi)= \int_{\mathbb{R}^2}k\phi dx$ for all $\phi$

(b). Let $u$ be a distribution on $\mathbb{R}^2$. Assume there exists a continuous function $h$ on $\mathbb{R}^2$ such that $(u,\Delta \phi) = \int_{\mathbb{R}^2}h\phi dx $ for all $\phi\in ...
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1answer
353 views

Calculating Inverse Fourier Transform

I can't quite get an inverse Fourier Transform to match up with a statement in my textbook. At one point, my textbook writes: "If $g$ is a function that is one on the interval $(- \pi, \pi]$ and ...
2
votes
2answers
112 views

Uniqueness of distribution with moments $M_n$ if $\limsup_{n\to\infty} \frac{1}{n}\sqrt[n]{M_n}$ finite

There's a theorem which states that the moments, i.e. $M_n = \mathbb{E}\left(X^n\right)$, of a distribution uniquely identify the distribution if $$ R := \left(\limsup_{n\to\infty} ...
2
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1answer
69 views

Showing that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$

I want to show that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$ $C^0$ is the space of continuous functions, and $H_{\text{loc}}^2(\mathbb{R}^2)$ the set of distributions $u\in ...
5
votes
2answers
495 views

A Fourier series exercise

Can anyone give me a hand with this exercise about Fourier series? Let $f(x)=-\log|2\sin(\frac{x}{2})|\,\,\,$ $0\lt|x|\leq\pi$ 1) Prove that f is integrable in $[-\pi,\pi]$. 2) Calculate the ...
1
vote
1answer
591 views

how to find absolute value for complex fraction

I have a Fourier transfer equation $H(jw) = \frac{jwL}{(jw)^2LC+jw\frac{L}{R}+1}$, and I need to find frequency to make $|H(jw)|$ is max. I know I should take the derivative of $|H(jw)|$ then find ...
5
votes
1answer
242 views

Highly Oscillating Integrals

I'd like to know the behavior of integrals of the form: $$ \int_0^1 f(x) \cos(k x) dx $$ as $ k \rightarrow \infty $ where f is a smooth function. It is easy to see, by expanding f in power series, ...
10
votes
4answers
414 views

Singular asymptotics of Gaussian integrals with periodic perturbations

At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$, $$ \int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
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0answers
107 views

Intervals where the function is similar to the Fourier series

$$f(x)=\left\{\begin{array}{l l} 0,\quad x \in [-L,0[\\ 1,\quad x \in [0,L] \end{array}\right.$$ I need to know in which intervals the sum of the Fourier series is "equal to the function $f(x)$". ...
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1answer
45 views

How to prove this Fourier question?

How to prove this Fourier question? I hope for a procedure in detail.
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0answers
184 views

Fourier Analysis of Prime Counting Function

I was thinking about the following: Denote $\pi(x)$ as the prime counting function such that: $$ \pi(x) = \#\text{ of prime numbers}\leq x $$ It is well known from the prime number theorem that $$ ...
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0answers
177 views

Problem 25 pg 95, Stein and Shakarchi: $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$.

Show that for any $\epsilon>0$, the function $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$. [Hint: $K_{\delta}(X) = e^{-\pi|x|^{2/\delta}} ...
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2answers
146 views

Problem #23 pg-94, Stein and Shakarchi

As an application of the Fourier transform, show that there does not exist a function $I\in L^1(R^d,m)$ such that $f*I = f$ for all $f\in L^1(R^d,m)$.
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votes
3answers
530 views

Fourier transform of a compactly supported function

In which space does the Fourier transform of a smooth compactly supported function $\phi$ lie? I would not say it lies in $\mathcal{S}$, heuristically as one can approximate the step function which is ...
1
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1answer
138 views

Are Trigonometric Functions Dense in $C^k(S^1)?$

Consider the functions $\{e^{2\pi i nx}\}_{n \in \mathbb{Z}}$ defined on the interval $[0,1].$ These are all smooth periodic functions (so functions on $S^1)$ and by the Stone-Weierstrass theorem ...
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0answers
40 views

Understanding the indices in a Fourier series

Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written $$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$ which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
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1answer
208 views

Only when the function is odd its Fourier coefficient of cosnx is equal to 0?

Only when the function is odd its Fourier coefficient of cosnx is equal to 0?
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0answers
96 views

Inversion of a Fourier Transform

I am told that the functions $f(x), g(x)$ and $h(x)$ satisfy: $\hat{f}(k) = \dfrac{\hat{h}(k)}{A+\hat{g}(k)} $, where $\hat{f}(k)$ is the Fourier Transform of $f(x)$ (likewise for $h(x)$ and $g(x)$ ...
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0answers
116 views

What are the properties of this kind of Fourier transform when using it to solve Wave equqtion with Robin boudary data?

I am facinated by the following problem (Which is found in Budak & Samarskii's Book Collection of Problems on Mathematical Physics, about on Page 53.): Solve the following problem by using ...
4
votes
1answer
2k views

Is deconvolution simply division in frequency domain?

Is it correct to say that deconvolution simply division in frequency domain? And that convolution in time domain is multiplication in frequency domain. And is it a convention to notate a function in ...
3
votes
2answers
120 views

Fourier Transform calculation

I am trying to calculate the Fourier Transform of $$f(x)=\exp(-\frac{|x|^2}{2}). $$ Thus, I am looking at the integral $$ \hat{f}(u)=\int_{\mathbb{R}^n} \exp(-\frac{|x|^2}{2}) \cdot \exp(ix\cdot u) ...
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1answer
432 views

Discrete Time Fourier Transform example: $x = [1 \; 2 \; 3 \; 4]^T \; \rightarrow \; X=?$

How do I find the Discrete Fourier Transform of the sequence below? $$ x = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$$ Show all steps.
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1answer
83 views

Why aren't these two question equal?

Firstly I doubt whether the 12 is right in Q1.If it is right,please give a proof. Secondly why (1) is not equal to (2) in Q2?