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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6
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1answer
44 views

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$, looking at $C_{(2)}[-1,1]$, with $L_2$ norm. I tried to look at a general polynomial $\sum_{i=0}^{98} ...
2
votes
1answer
27 views

On a property of $(Mf)^{\delta}$

For each positive $C$, define a set $$A_C=\left\{g\ge0: \frac{1}{|I|}\int_Ig\le C\inf_{x\in I}g(x) \text{ for any interval } I\right\}$$ In other words, elements in $A_C$ are non-negative functions ...
0
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0answers
15 views

Fourier transforms of some homogeneous functions

In $2D$, what is the Fourier transform (in the sense of distributions) of functions of the form $x_i/|x|^2, 1/|x|, x_i/|x|, x_i x_j /|x|^2$, and so on? Here, $i = 1,2$. They are homogeneous and ...
2
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0answers
24 views

Relation between representation of a number in an integer base and Fourier series representation of a periodic signal

I am not a Mathematician - am just a software developer though I did some "Math" back in the day as part of my undergrad studies millions of years ago. Recently I had to revisit Fourier analysis of ...
2
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2answers
40 views

Inequalities in proof of Bernstein-type lemmas

I'm working through the proof of Lemma 2.1 in Bahouri-Chemin-Danchin's Fourier Analysis and Nonlinear PDE. The place where I'm stuck boils down to (I think) proving the following inequality: ...
4
votes
3answers
106 views

Develop $x\sin(x)$ into a specific series

I have to find a series that in $0\leq x\leq \pi$: $$x\sin(x) = \sum_{n=0}^{\infty} a_n\sin(2nx) $$ It seems to be impossible because on $x=\pi/2$ we get $ \pi/2 = 0$. However I tried to do it (as ...
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0answers
34 views

Complex measure integral vanishes

I have a complex measure $\mu$ on $(-\pi,\pi)$ with $\mu((-\pi,\pi))=0$ and a continuous function $g$ on $(-\pi,\pi)$ with the property $\hat g(k)=0$ for $k=-1,-2,\dots$ (Fourier coefficients w.r.t ...
0
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0answers
25 views

Strong period of self-convolution of a strongly periodic cyclic function

Let $f : \mathbb{Z}_N \to F$ be a function with a $F$ a field. We say that $f$ has maximum period if the smallest positive integer $r$ (with $r \mid N$) such that $f(j) = f(j + r)$ for all $j \in ...
1
vote
1answer
66 views

Fourier-Transform of $\frac{x^{\alpha}}{1+x^2}$

Does anybody know a way to compute the Fourier-transform $F(y)=\int_{-\infty}^{\infty} e^{-i xy} f(x) dx$ or Cosine-Transform $G(y)=\int_{0}^{\infty} cos(xy) f(x) dx$ of the function $$ ...
3
votes
1answer
45 views

Inverse Fourier Transform

I want to find the inverse Fourier transform of the function $g(k) = e^{−a|k|}$, $a > 0, −∞ < k < ∞$. Now I proceed using $\mathcal F^{-1} [g(k)] = \int_{−∞}^∞ e^{ikx}g(k)$. I chose to split ...
2
votes
0answers
36 views

Fourier transform of distribution solution

Let $f(x)=2$ for all $x$. What is the Fourier transform of $f$? This is my solution but there are some steps I don't fully understand, I took it from an example just to get through the rest of the ...
2
votes
1answer
33 views

Problems in understanding Lacey's proof of Carleson's theorem.

I am new to the Stackexchange community so do let me know if I can improve my question in any way. Right, I have just started reading Michael Lacey's proof of Carleson's theorem ...
0
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0answers
20 views

Fourier transform of a two variables function with respect to one variable.

Suppose I have function $f $ on $\mathbb R^2$ and I want to compute the fourier transform of $f $ relative to first variable, Is it true that for every $ t\in\mathbb R $ $\hat f ( \xi,t)=\int f ...
4
votes
1answer
37 views

A Schwartz function is identically zero on $\mathbb R^2$ if its integral on every line in the plane is zero

If the integral of a Schwartz function is zero on every line in the plane then it is zero. I think maybe Fourier transform in one variable is useful. But I can't success. Is there any hint to ...
0
votes
0answers
18 views

Taking the square of an image in Fourier domain, why not square of real part?

In my quest to understand Math during the Christmas holidays I'm working on Fourier transforms today. I understand that a single point in Fourier space corresponds to line in normal 2D image space. ...
0
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0answers
11 views

How to estimate the spectrum density matrix given a basis?

I have a basis and I will use $f(\omega, t)$ denote it, in which, $\omega$ is the discrete frequency parameter and $t$ is the discrete time parameter. Also, I have a sample with size $N$ and time ...
3
votes
1answer
97 views

Inverse short time Fourier transform

The short time Fourier transform $S: L^2(\mathbb{R})^2 \rightarrow L^2(\mathbb{R}^2)$ can be defined as $$S(g,f)(a,b):=\int_{\mathbb{R}}f(x) \overline{g(x-a)} e^{-i b x} dx.$$ Now a natural question ...
0
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0answers
18 views

Is it possible to do effectively irrational-interval sampling of a continuous signal?

Suppose there is a real-valued $f(t)$, with $t$ being time. And one wishes to sample at interval of $\pi$, for example. Perfect irrational-interval sampling is not possible, but is there a way to do ...
8
votes
1answer
60 views

Almost-identity: $[\int_0^\infty{\rm d}x-\sum_{x=1}^\infty] \prod_{k=0}^N\text{sinc}\left(\frac{x}{2k+1}\right) = \frac{1}{2}$

Show that the identity $$\int_0^\infty \prod_{k=0}^N \text{sinc}\left(\frac{x}{2k+1}\right)\,{\rm d}x - \sum_{n=1}^\infty \prod_{k=0}^N \text{sinc}\left(\frac{n}{2k+1}\right) = \frac{1}{2}$$ ...
1
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0answers
40 views

Sampling, Fourier Transform, and Discrete Fourier Transform

The Fourier Transform inverse Fourier Transform and are defined as: $$F(k) = \int_{-\infty}^\infty f(x)e^{-2\pi i k x}dx \\ f(x) = \int_{-\infty}^\infty F(k)e^{2\pi i k x}dk$$ The Discrete Fourier ...
0
votes
1answer
28 views

is there any good way to figure out number of fourier series frequencies of some signal?

Suppose you have $f(t)$, but you do not know the exact function and can only measure $f(t)$ at certain time. Assume $f(t)$ is complex-valued with $t$ being "time." One wishes to find out the number ...
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0answers
24 views

What is the analogue of $f(x)=e^{-x^2}$ on the torus? What about its Fourier transform?

Let $f:\mathbb R \to \mathbb R$ such that $f(x)=e^{-x^2} \ (x\in \mathbb R).$ We know that $f, \hat{f} \in L^{1}(\mathbb R).$ My Question is: What is the natural analogue function of ...
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0answers
16 views

Fourier transform of wave function and momentum of particle

Let $$\Psi\left ( x,t \right )$$ represents the wave function of a particle at some position x and time t. $$\Psi\left ( x,t \right )=\frac{1}{\pi \sqrt{2a}}\int_{-\infty}^{\infty}\phi\left ( k ...
0
votes
1answer
20 views

Integral $\int_{0}^{T} e^{-jnwt}dt$

I'm representing some function using complex Fourier series and I have to solve this integral: $\int_{0}^{T} e^{-jnwt}dt$, where $w=\frac{2\pi}{T}$ I got this: $\int_{0}^{T} ...
0
votes
1answer
36 views

Energy of a Signal

A signal is given by $$x(t)=\begin{cases} e^{-t} &t\geq 0\\ 0 &t < 0 \\\end{cases}$$ Find the Fourier transform of the signal. $$X(\omega)=\int_{0}^{\infty}e^{-t}e^{-j\omega t} dt$$ ...
2
votes
1answer
32 views

Fourier transform and differentiability

Let $f $ be $L^1 (R) $, and $x\mathcal {F}(f)(x) $ be also $L^1 (R) $. Prove that there exists $g $ differentiable which holds $f=g $ almost everywhere. I know that has some relation with the fact ...
1
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1answer
14 views

Relationship between short-time and large-frequency asymptotics in Fourier transform

I am trying to understand how the short-time behaviour of a function $f(t)$ influences the large-frequency asymptotics of its Fourier transform $g(\omega)=\mathcal{F}[f(t)](\omega)\equiv ...
0
votes
1answer
29 views

What is Fourier transform of $g(y)=e^{-\pi y^2 +2\pi yx}$ for all $y\in \mathbb R.$?

Let $x\in \mathbb R.$ Define $g:\mathbb R\to \mathbb R$ as $g(y)=e^{-\pi y^2 +2\pi yx}$ for all $y\in \mathbb R.$ My Question is: What is the Fourier transform of $g$? In other word, how to ...
1
vote
0answers
20 views

Inverse Fourier Transform of (e^w)*F(w)

If the inverse Fourier Transform of F(w) is f(t), what is the inverse Fourier Transform of (e^w)*F(w) ? My best guess is that I should expand e^w into a power series and use the fact that the inverse ...
2
votes
0answers
39 views

Change from Fourier Space to Real Space

I have a function in 3D fourier Space $$g(\textbf {k})=\frac{\hat{k}_i}{\hat{k_j}}f(\textbf {k}),$$ where $\hat{\alpha}$ is a fixed vector and $i$ and $j$ are the components of the relevant vector, ...
1
vote
1answer
29 views

Showing that complicated Fourier integral satisfies heat equation

I'm doing a chapter on Fourier analysis and I got a rather involved problem. Given the function $u$, \begin{align*} u = \frac1{\sqrt{\pi t}} \int_{-\infty}^\infty ...
0
votes
1answer
39 views

Is $\|\widehat{f}\|_{L^{p}(\mathbb{R}^d)}$ comparable to $\|\widehat{f}\|_{L^{p}(B_{1/R})}$?

Suppose $f$ is a smooth function compactly supported in some ball of radius $R$. Is $\|\widehat{f}\|_{L^{p}(\mathbb{R}^d)}$ comparable to $\|\widehat{f}\|_{L^{p}(B_{1/R})}$ where $B_{1/R}$ is any ball ...
1
vote
1answer
90 views

Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$ using Fourier series

Consider the function $f(x) = \frac{x}{2}$, defined over the interval $[0, 2\pi]$. Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$.
0
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0answers
12 views

determine spacing in dirac comb

Lets imagine the sum of two not equally spaced dirac combs. Is there a transformation to get the spacing of the signal? To further elaborate my point. If the signal would be a superposition of to ...
0
votes
2answers
38 views

Find spectrum of $f(t-t_0)\sin(\omega_0(t-t_0))$

If functions $f(t)$ and $F(j\omega)$ form a Fourier Transform pair, how do I find the spectrum of the function $$f(t-t_0)\sin(\omega_0(t-t_0))?$$
1
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0answers
16 views

average order of basic trigonometric functions

In connection to a problem in a Fourier Analysis book, I had to calculate the average order of some functions. Then I came across this very elementary yet very interesting observations which follow: ...
1
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2answers
30 views

Showing $\hat{\tilde{f}}=\tilde{\hat{f}}$ where $\hat{f}$ is the Fourier transform, and $\tilde{f}(x) = f(-x)$

I'm trying to prove that $\hat{\tilde{f}}=\tilde{\hat{f}}$ for any integrable function $f$, where $\hat{f}$ denotes the Fourier transform of $f$ and $\tilde{f}$ denotes the mapping $f(x)\to f(-x)$, ...
1
vote
1answer
40 views

Fourier Transform for PDE $u_{xy}$

Using the Fourier transform, $\mathscr{F}\{\cdot\} = \hat{f}(\cdot)$, may I transform the $u_{xy}$ where $u=u(x,y,t)$, $x,y \in (-\infty, \infty)$ and $t \in (0,1]$? The initial condition is ...
0
votes
1answer
33 views

$u(x, t)$ of \begin{equation} \begin{cases} u_t = k u_{xx} + u \\ u(x, 0) = f(x) \end{cases} \end{equation}

How do I use a Fourier transform to find a formula for the solution $u(x, t)$ of \begin{equation} \begin{cases} u_t = k u_{xx} + u \\ u(x, 0) = f(x) \end{cases} \end{equation} ...
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0answers
42 views

Leibniz formula using Fourier Series

I have to show the Leibniz formula i.e $$\pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...$$ and I have to do so using $f(x) = x/2$ on the interval $[0,2\pi]$ for this function being $2\pi$ periodic. It is clear ...
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0answers
22 views

Proof of an inequality with using maximal operator

I want to prove an inequality such that $$ \int_{B}|f(y)|dy\leq |B|^{1-\frac{1}{p}}\|f\|_{L^p(B)}, $$ where $B\subset\mathbb{R}^n$ is a ball, $p>1$ and ...
2
votes
2answers
38 views

Why does $\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$

Why does $$\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$$ This is used in a derivation of the Fourier coefficients. I see why ...
0
votes
1answer
24 views

solving integral equation by fourier transform

this is fredholm type of integral equation that im currently triying to solve $$\int_{-\infty}^{\infty}e^{-(t-x)^2}\phi(t)dt=e^{-x^2/8}$$ the question is that how to derive $\phi(t)$ i tried ...
2
votes
1answer
46 views

Ways to generate triangle wave function.

I recently when searching for parameters on a unit cube in $\mathbb{R}^9$ (we all have our more or less peculiar hobbies, don't we?) found a practical reason to implement a triangle wave function ...
0
votes
0answers
27 views

Question related to decay of Fourier transform and smoothness

Suppose $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ Let $$g(x) = \frac ...
2
votes
1answer
23 views

Convolution Problem

while working on a signal processing problem i've reached to the following: So my aproach was: Am I doing something wrong? Is it valid Y(f)=[X(f) x H(f)]*W(f)=X(f) x [H(f)*W(f)] If you could ...
0
votes
0answers
50 views

An exercise from stein's fourier analysis

I'm trying to solve Exercise 20 of Chapter 5 of Fourier Analysis by Stein. The problem is as follows: Suppose $f$ is of moderate decrease and that its Fourier transform $\hat{f}$ is supported in ...
1
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0answers
30 views

Fast evaluation of an integral convolution with an “expanding kernel”

Suppose I have a 1-D integral convolution transform like this: $$ g(x) = \int_{-\infty}^{+\infty} dy\, f(y)\, K(x-y). \qquad (1) $$ Say the kernel $K(x)$ is a known analytic function, and say we have ...
1
vote
1answer
29 views

Fourier transfrom of $te^{-2t}$ for $t\geq 0$

I'm trying to calculate the fourier transform of Fourier transfrom of $te^{-2t}$ for $t\geq0$. I'm given the hint that the fourier transform of a function that has the form $e^{-at}f(t)$ is $F(a + ...
0
votes
1answer
19 views

What is the effect on a Fourier transform of multiplying in the time domain by a constant?

The Hilbert transform is given by the following: $$ \mathscr H[a(t)] =\frac 1π \times a(t) *\frac 1t $$ Now, I'm trying to do envelope detection on an audio signal and I'm hoping you math folks can ...