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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How to prove this Fourier question?

How to prove this Fourier question? I hope for a procedure in detail.
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247 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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2answers
160 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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847 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 ...
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1answer
177 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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45 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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295 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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123 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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122 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 ...
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1answer
3k 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 ...
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2answers
129 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
450 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
86 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?
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1answer
73 views

I can't understand the last second step.

It is known that $f(x)=\sum_{n=-\infty}^{\infty}c_{n}e^{inx}$, with $c_{n}:=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\:dx$, for $n\in\mathbb{Z}$. To prove: ...
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1answer
162 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
3
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1answer
78 views

Fourier Transform is onto $L^2(\mathbb{R})$

In Introduction to Fourier Analysis on Euclidean Spaces, the authors explore the $L^2$ extension of the Fourier transform and argue that it is onto $L^2(\mathbb{R})$ but I can't follow their ...
3
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3answers
127 views

Klein Bottle discrete harmonics?

Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus. ...
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1answer
122 views

number of zeros of the superposition/interference of sine oscillations

There is a tricky problem to solve and we ask for your kind help. In a rather simple version of the problem, we have a set of oscillations of type $f(x)=\sin (\omega x + \phi)$ with different phases ...
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2answers
39 views

nasty exponentials

While trying to find the fourier transform of $\Large \frac{1}{1 + x^4} $, using the definition and the residue theorem has required me to evaluate nasty looking expressions like $$\large \rm ...
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2answers
92 views

Show existence of continuous functions $f$ with $f''=\delta_0-\delta_1$

Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$ (a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with ...
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1answer
60 views

Show that Fourier coefficients approach zero uniformly

Let $f(t)$, $g(t)$ be piecewise continuous functons on $[-\pi,\pi]$, periodically continued on $\mathbb R$. I want to show that $$ a_n(x) = \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x+t)g(t) ...
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1answer
214 views

Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform

This is a question from one of the past papers of my university which I am unable to do. I am not being able to do question 2 from below. Let $f(x)= a^2-x^2 \,\,\,\,\, |x|<a ...
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2answers
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Calculate the Fourier transform of $b(x) =\frac{1}{x^2 +a^2}$

I need help to calculate the Fourier transform of this funcion $$b(x) =\frac{1}{x^2 +a^2}\,,\qquad a > 0$$ Thanks.
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1answer
78 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
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Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
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0answers
87 views

Fast Fourier Transform of 2D axisymmetric geometry

I need to compute the FFT of a signal in 2D axisymmetric geometry. The signal consists of a snapshot in time of a laser beam for which I have values in z (direction of propagation) and r (from 0 to ...
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2answers
2k views

Solving an integral equation using the Fourier transform

I have to solve the equation $\int_0^{\infty} f(x) \cos{(\alpha x)}\, dx=\frac{\sin{\alpha }}{\alpha}$ Using fourier transform. I know this is half of the usual fourier cosine transform, and so ...
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1answer
115 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
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1answer
77 views

Fourier transforms - don't understand this concept!!! Please help me on this

I have two Fourier transforms to solve, but the problem is that a I have a characteristic bijection or some etching that I don't know what it is and I don't know how to solve this... Please help ...
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1answer
73 views

Fourier transform parameter

I'm currently working on the problem: If $\hat{f}(k)$ is the complex Fourier Transform of the function $f(x)$ and $a$ is a real constant with $a>0$, show that the complex Fourier Transform of ...
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Sum over cosines = dirac delta - how to get the coefficients?

Given this formula: $$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$ Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$? I googled and searched all kinds of ...
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30 views

Complex Fourier series of a function [duplicate]

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
3
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1answer
454 views

Complex Fourier series

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
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2answers
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Fourier Series of $f(x) = x$

I am having trouble finding the complex Fourier series of $f(x) = x$ and using that complex series to find 1)the real Fourier series of $f(x)$ and 2) the complex and real Fourier series of $h(x) = ...
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1answer
136 views

Unclear on relationship between different dimensionalities of Fourier transform

This is probably a silly question, but it's one that's directly relevant to a project of mine and I figured this was the place to go. I have some objects that contain a 1d and a 2d array of double ...
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2answers
151 views

Is $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$ if $g$ is a Sobolev function on the real line?

If we are given a function $g\in W_2^k(\mathbb{R})$ (even consider $k=1$ for simplicity), then is it true or not that $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$? That is, do we have ...
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1answer
577 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
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1answer
52 views

A function whose derivatives always have a convergent fourier series

I am looking for a solid example that such a function that its derivatives can always be found by taking derivatives component-wisely in its Fourier series. A function with finitely many Fourier terms ...
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2answers
358 views

Fourier transform of $(1-\cos(tx))/x^2$

I am trying to compute the Fourier transform of $f(x) = \frac{1-\cos(tx)}{x^2}$, $(t > 0)$ directly. I tried contour integration, and could not seem to get it to work. So, I am wondering if it can ...
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3answers
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How to reperesent $\sin^{4}(x)$ byFourier series? [closed]

how to represent $\sin^{4}(x)$ by Fourier series? Obviously,$\sin^{4}(x)$ is an even function, so $b_n=0$. How can i get $a_n$ ?
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2answers
158 views

How to solve this equation by Fourier series?

$$ y''+3y=\sin ^4 x ,\quad y=\frac{1}{8} +\frac{\cos2x}{2}-\frac{\cos4x}{104}.$$ Now the text book states the solution, but I don't know the process of solving this equation. I need your help!
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1answer
66 views

I need to find the Fourier transform of the following function

I need to find the Fourier transform of $e^{-2 \pi |x|}$. Normally, I can do something like this, but the absolute value is kind of confusing me.
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1answer
321 views

Show that f is a polynomial

Suppose $f$ is an entire function on $\mathbb{C}^n$ that satisfies for every $\epsilon>0$ a growth-condition $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon | \text{Im}\,z|}$$ ...
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428 views

Upper bound on truncation error of a fourier series approximation of a pdf?

Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation: $$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$ ...
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1answer
62 views

Flipping and convolution theorem

Could you provide the mathmatical proof that multiplication in Fourier domain is only convolution , when the flipping (of one of the signals/functions) occurs. So, that multiplication is not ...
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1answer
560 views

Fourier Transform on Infinite Strip Poisson Equation

Im trying to solve the following Poisson equation: $$u_{xx} + u_{yy} = \exp(-x^2)\ \text{for}\ x \in (-\infty, \infty)\ \text{and}\ y \in (0,1)$$ $$u(x,0) = 0,\ u(x,1) = 0$$ $$u(x,y) \to 0\ ...
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1answer
81 views

Understanding fourier notation $F(\partial_x)$

Can somebody please help me understand some of the notion in the equations below, taken from a published paper on image de-blurring. I have an energy $E(H)$ defined over an image $H$, a point-spread ...
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55 views

What is the relationship between singularities for complex times and high frequency asymptotics?

As said in a paper I am reading on p 2677 in the text directly above FIG3, this should be a standard result about Fourier transforms of analytic functions. In the paper the authors use these methodes ...
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1answer
201 views

Pointwise convergence of double Fourier series

I'm looking for theorems that deal with the pointwise convergence of double Fourier series expansions for a special class of functions. Let $D \subset [-\pi, +\pi]^2$ be an arbitrary set of finite ...
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2answers
86 views

Computing the Fourier transform of a certain function

Let $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ be the function defined by $f(x)=(1+|x|^2)^{-1}$. Problem: How can I find $\hat{f}$ by direct computation? Remark: This is exercise 8.5 in Rudin's ...