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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Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
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1answer
23 views

Fourier Transform Properties Proof

If: $y(t) = x(t)*h(t)$ and $g(t) = x(9t)*h(9t)$ (Where * is convolution) How can I use properties of the Fourier transform to show: $g(t) = Ay(Bt)$ and find constants? I think A should be $1/81$ ...
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0answers
21 views

Vanishing Fourier terms

Which Fourier coefficients vanish for a periodic function $ f(\theta) $ of period $ 2\pi $ satisfying $ f(\theta) = f(\pi − \theta) $? What about $ f(\theta) = - f(\pi − \theta) $ 􏰖Hint: ...
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14 views

Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs: $$\hat{L}u(x,t)=0$$ with the constraint $|u\big(x,t\big)|^2=|u\big(f(t)\cdot x + g(t),0\big)|^2$. During evolution, the ...
2
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0answers
24 views

How to find the global minimum of a function from its Fourier transformed function.

i.e Can $\min{f(t)}$ be expressed by $F(\omega)$? I have a series of data in frequency space. I can do discrete Fourier transform to time space to find its minimum. But I am wondering if there is a ...
0
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1answer
26 views

Fast Fourier Transformation: inverse transform of the product of polynomials

I have managed to implement and understand most of the Fast Fourier Transformation. However, I have one last question. If one has two polynomials, say $A(x)$ and $B(x)$, and one computes DFT of ...
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1answer
31 views

About composition of Holder functions.

Let $f,g$ be Holder continuous functions with respective exponents $\alpha, \beta \in (0,1)$. More precisely $f \in C^{\alpha}(\mathbb{R}^n;\mathbb{R}^n)$, $g\in C^{\beta}(\mathbb{R}^n,\mathbb{R})$. ...
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34 views

Fast Fourier Transformation, explanation of $A(x)=A_0(x^2)+x A_1(x^2)$

I was looking at the Fast-Fourier Transformation today, on this site [if you cannot read Russian, simply use Google Translate, which is what I am doing right now]. http://e-maxx.ru/algo/fft_multiply ...
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2answers
56 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
2
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1answer
28 views

Fourier transform of a Laplace transform

Is there an easy way to find the Fourier transform of a Laplace transform of function? $$ F[L[f(t)]_{s}] $$ Where my $f(t)$ is $\sqrt{t}$. However, Before finding the Fourier transform I do the ...
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27 views
+50

Equidistribution of $an^\sigma$

I am stuck at an exercise in Stein's book: Fourier Analysis, and it's exercise 8 in chapter 4. Show that for any $a\ne0$, and $\sigma$ with $0<\sigma<1$, the sequence $an^\sigma$ is ...
3
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1answer
65 views

Functions for which $\mathcal{F}g = f \ast f$

Suppose one is given $f \in L^{2}(\mathbb{R})$, my question is whether or not there exists a $g \in L^{1}(\mathbb{R})$ such that $f \ast f = \mathcal{F}g$ where $\mathcal{F}$ is the Fourier transform. ...
4
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0answers
36 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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0answers
150 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
0
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1answer
41 views

Why would the discrete fourier transform “see” signals like this? What is the origin of spectral leakage?

The discrete fourier transform of $x = (x_{0},\dots,x_{N-1})$ is defined as $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N}$ where $\omega^{kn}_{N} = e^{-2\pi ik/N}$ and ...
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1answer
23 views

Is it true that, $x\rho(x/t)\in H^{s}$ for $\rho\in \mathcal{D}(\mathbb R)$ and $s>3/2$?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support; and the Sobolev space $$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} ...
0
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2answers
104 views

$f\in L^{\infty}(\mathbb{R})$ as Fourier transform

i need to know if one can view a function $f\in L^{\infty}(\mathbb{R})$ as a Fourier transform of a certain function, say g? If the answer is positive please state the proof, or help me find one. ...
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4answers
39 views

How to determine the periods of a periodic function?

I am aware of the other similar questions but was not able to figure out what I want to know from those question thus posting it here. Given a periodic function $f(x)=sin(x)$, Why is the period ...
2
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1answer
40 views

Embedding of weighted Lebesgue space on Fourier side into $C^\infty$

Given $g$, a continuous real-valued function defined in $\mathbb R^n$, $K^g:=\{u:e^g \hat{u}\in L^2(\mathbb R^n,(1/2 \pi)^nd \xi)\}$. Thus $K^g$ can be viewed as a Hilbert space with natural norm ...
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0answers
24 views

Intuition about generating functions

I am trying to gain some intuition about moment generating functions. In particular, for a random variable $X$, we have $$ \newcommand{\E}[1]{\mathbf{E}\!\left[#1\right]} M_X(t) = \E{e^{Xt}} = ...
1
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1answer
31 views

Does the Fourier coefficients of a function $f\in H^1(0,L)$ (the first order Sobolev space) are absolutely summable?

My precise question: Let $f\in H^1(0,L)$ and let $\{f_n\}$ be its Fourier sine series coefficients on $(0,L)$, is it true that $\{f_n\}\in l^1$, i.e. $$\sum_{n}|f_n|< \infty .$$ Thanks
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34 views

Fourier Inversion transform

Is what i have highlighted in green a typo? Should it be $\pi e^{-|\xi|}$?
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1answer
25 views

Fourier transform of a triangular pulse

So I've been practicing some fourier transform questions and stumbled on this one; To start off, i defined the fourier transform for this function by taking integral from -tau to 0 and 0 to tau as ...
0
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0answers
20 views

pFourier Transform using symbolic toolbox

Hi i need to use the symbolic toolbox from matlab and solve the following by fourier transform: $\int^{\infty}_{-\infty}\frac{\sin(at)\sin(bt)}{t^2}dt\text{ }\text{}\text{}$ So far the algo i came ...
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1answer
23 views

Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
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0answers
37 views

An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
0
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1answer
18 views

characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
3
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1answer
37 views

What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
3
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1answer
35 views

Poles of Fourier transform

Let $f\in L_2(\mathbb R_+)$ and consider its Fourier transform $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ Is it true that analytic continuation of $F(\zeta)$ has at most finitely many poles in a ...
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1answer
25 views

Paley-Wiener theorem for a sector $\{\zeta:-\epsilon<\arg(\zeta)<\pi+\epsilon\}$

One of the variations of the Paley-Wiener theorem yields: If $f\in L^2(\mathbb R_+)$, then the Fourier transform $F$, defined by $$F(\zeta)=\int_0^\infty f(x)e^{ix\zeta}dx$$ is a holomorphic function ...
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30 views

Can integral transforms be viewed as change of basis formulas?

Forgive any lack of rigor, this question is kinda all over the place. If you have a set $B $ of $ N $ basis functions $ g_0(t), g_1(t), g_2(t), \dots, g_{N-1}(t) $ which are orthogonal over $[t_1, ...
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2answers
35 views

Is the convolution operation some kind of group operation?

I'm just curious but will the convolution operation be any sort of group operation? A motivating example would be to see that the natural exponential family of distribution functions are closed under ...
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0answers
28 views

Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
3
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1answer
93 views
+100

Amount of sound near a specific frequency at a specific time

I have a sound signal, sampled at 48000 hz. Now I want to know 'how much sound' there is of a specific frequency (or near that frequency) at a specific time. For example, I want to know at 10s how ...
6
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62 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
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3answers
32 views

What are the concepts that I need to understand before studying Fourier Analysis?

Background ( Long Story Short ) : For some reasons, I am taking a class in my university that focus on Fourier Analysis Laplace Transform, and Partial Diffiential Equations Problem : I have done ...
0
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1answer
25 views

Help for understanding Danielson-Lanczos lemma

The Danielson-Lanczos lemma is the basis for fast Fourier transform algorithms. Now, I do understand this step $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N} = \sum_{n=0}^{(N/2)-1} ...
1
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1answer
27 views

The meaning of the connection between power spectral density and auto correlation

I know that if we have a signal $x(t)$, then its Fourier transform would be the signal in the frequency space, which I understand to be how much of each frequency exists in the x(t) signal. $ ...
0
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1answer
56 views

Looking for a nice expression of these functions in terms of trig functions

I have come across three sinusoidal functions f1, f2, and f3 which, up to scaling and translation, are very close to each other. When normalized and plotted together, they are hard to tell apart. ...
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1answer
25 views

Square Summable functions

Can somebody please help me understand the notion of square summable functions intuitively?? I have been self studying Hilbert Spaces and Fourier Transform for DSP. Any help is appreciated. Thanks.
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46 views

Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$ \check{\hat{f}}=\hat{\check{f}}, $$ where $$ \hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x $$ and ...
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1answer
34 views

Can we expect to find $r,$ large enough, so, $\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C (1+y^{2})^{s} $ for all $y\in \mathbb R$?

Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we expect to find $r$ large enough, so that ...
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0answers
15 views

What is the equation for an anisotropic Hanning window (cosine wave) in two or three dimensions?

I do not exactly know how to ask this question, so I will explain myself thoroughly. I am really stuck on this one, and it is crucial for my research, so if anyone has any ideas on where I may find ...
3
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0answers
31 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
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1answer
33 views

What is the Fourier series of $\frac1T\sum^{\infty}_{m=-\infty}\delta(f-\frac mT)$?

As the title mentioned, I've not known exactly about Fourier series and when I was reading an digital communication textbook, I wondered about below equation derivation of Fourier series like ...
4
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4answers
78 views

a question how to prove:$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos(nx)}\over {n}}=\ln(2\cos(x/2))$

I found a complicated question in my textbook, I can't solve it? How to prove $$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos nx}\over {n}}=\ln(2\cos(x/2))$$ where $x\in(-\pi,\pi)$. My tried method: I tried ...
1
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0answers
28 views

Is Sobolev space $H^{s}(\mathbb R),$ for $s>\frac{1}{2},$ closed under point wise multiplication? [duplicate]

We note that, $L^{2}(\mathbb R)$ is not closed under point wise multiplication. Let $s>\frac{1}{2};$ and we define Sobolev space, as follows: $H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb ...
2
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1answer
56 views

Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$

Problem: Let $Lf =-((1-x^{2})f')'$ be the Legendre differential operator defined on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions on $(-1,1)$ for which $f, Lf \in ...
0
votes
1answer
63 views

inverse of a circulant matrix

If c(x) is a circulant matrix of a vector x, is it always true that $c(x)^{-1}=\mathcal{F}^{-1}(\mathcal{F}(1/x))$? How can we prove it? In this case $c(x)^{-1}$ is the inverse of the circulant ...
2
votes
0answers
36 views

Decay of the Fourier transform of the surface measure of the sphere via uncertainty

I'm working through Tao's Recent Progress on the Restriction Conjecture notes (http://arxiv.org/abs/math/0311181). Currently, I'm working on problem 2.4, which will eventually allow us to compute the ...