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).

learn more… | top users | synonyms

0
votes
0answers
27 views

Given Fourier coefficients of a function , find the function

Given these Fourier coefficients: $$ X[k]=\begin{cases} 1 & \text{, k even}\\ 2 & \text{, k odd}\\ \end{cases} $$ I want to find the analytical expression for the function. What i tried was ...
0
votes
1answer
37 views

Dilation of Fourier transform

Let $f\in \mathcal{S}(\mathbb{R}).$ The Fourier transform of $f$ is defined by $\hat{f}(w) := \int_{-\infty}^\infty f(x) e^{-2\pi i x w} dx$. We use the notation $f(x) \longrightarrow \hat{f}(w)$ to ...
0
votes
0answers
21 views

Fourier transform of Si[$x^2 + y^2$]; Energy integrals involving sin integral functions

Problem Statement I'm trying to prove( or disprove ) the following identity \begin{equation} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\Big[\text{Si}[x_1^2 + y^2]- \text{Si}[x_2^2 + y^...
0
votes
1answer
24 views

Sequence $\{ e^{2 \pi i n x}\}_{n\in\mathbb N}$ and Paley-Wiener space $PW(0,1)$.

Let us consider the Paley-Wiener space: $$PW(0,1):=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset (0,1) \}.$$ Let us consider $\{ e^{2 \pi i n x}\}_{n\in\mathbb N}$, for $x\...
0
votes
1answer
15 views

Stuck finding inverse Fourier transform.

I have the equation $u_t - u_{xx} = f(x,t),\; x\in\mathbb{R},\;t>0$, with the initial condition $u(x,0) = 0$. I think I see where this is going but I want to make sure I'm not going in the wrong ...
1
vote
0answers
36 views

Find the function $f(x)$ by using its fourier coefficient

It is easy to find the fourier coefficient and fourier expansion of $f(x)$ function. But I want solve the inverse problem How to find the function $f(x)$, if I know its fourier coefficient (or ...
2
votes
1answer
40 views

Proving $\|[b,T](f)\|_{p}\le C\|b\|_{BMO(\mathbb{R}^{n})}\|f\|_{p}$ using the Fefferman-Stein inequality

Let $1<p<\infty$ and $1<r<\infty$ and let $\mathcal{M}_{r}(g)$ denote $\mathcal{M}(|g|^{r})^{\frac{1}{r}}$, where $\mathcal{M}$ is the Hardy-Littlewood maximal function. Also let $T\in ...
0
votes
0answers
19 views

Class of Functions , that admit Fourier transforms

For which class of functions/distributions is it sensible to take a Fourier transform ?
0
votes
0answers
112 views

Several questions about the proof of this lemma

I am reading this paper. I have Several questions about the proof of the lemma 3.3. How to Prove 1,2,3,4?
1
vote
1answer
44 views

Integration and differentiation of Fourier series

I am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose that ...
1
vote
0answers
24 views

Fourier Series and Fourier Transform confusion.

I dont understand the following paragraph after the proof. In particular, how does that theorem above give us that the Fourier transform maps $L^2$ onto $l^2$? all that theorem says is that this set ...
0
votes
0answers
9 views

Deconvolution by disks

I have a function $$f(x,y) = \begin{cases} 1, \|(x,y)-p\| < r\\0, \|(x,y)-p\|\geq r\end{cases}$$ where $p$ is some unknown point in $[0,1]^2$; i.e. $f$ is the characteristic function of some disk ...
0
votes
0answers
30 views

Dirac function expansion

In my book it is said that Dirac function $\delta(\tau)$ can be expanded as: $$ \delta(\tau)=(\beta \hbar)^{-1}\sum_{n \in even} e^{-i\omega_n\tau} $$ where $\omega_n=\frac{n\pi}{\beta\hbar}$, and $\...
0
votes
2answers
25 views

Gaussian is a rapidly decreasing function.

Definition of rapidly decreasing function $$\sup_{x\in\mathbb{R}} |x|^k |f^{(l)}(x)| < \infty$$ for every $k,l\ge 0$. Given the Gaussian function $f(x) = e^{-x^2}$, I know that its derivatives ...
2
votes
1answer
31 views

How to get from $\sum_{n=0}^N (a_n \cos{nx} + b_n \sin{nx})$ to $\sum_{-N}^{N} c_n e^{inx}$?

I'm currently reading Baby Rudin, and I'm in the section of Chapter 8 that covers Fourier series. There is one line that I just can't figure out for the life of me, and I can't find anything online ...
1
vote
1answer
17 views

What if we define function of moderate decrease as function satisfying $|f|\le\frac{A}{x^2}$

In Stein's Fourier Analysis, he defines the a continuous function $f$ as of moderate decrease if there exists $A>0$ such that $$|f(x)| \le \frac{A}{1+x^2} \forall x\in\mathbb{R}$$ I am wondering ...
1
vote
1answer
39 views

What is wavelet tranform in simple words?

I have read wiki and other sources and have still problem understanding the wavelet transform. What is the basic idea in simple words? Does the Fourier uncertainty hold for wavelet transform?
0
votes
0answers
26 views

Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ (...
0
votes
0answers
12 views

upsampling and plotting a signal in matlab

I want to upsample by 5 a signal in frequency domain, and then plot(stem) it. I figured how to upsample, Fk=(1/5)*upsample(ak_new,5) now this creates a vector ...
0
votes
0answers
6 views

How can I obtain the inverse transform?

The inverse Fourier transform is defined as: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$ I can't get an inverse Fourier Transform to Q1: $...
1
vote
0answers
21 views

Uniform convergence of Fourier series given certain conditions

If $f$ is a continuous, $a$-periodic and piecewise differentiable function on $[0,a]$ with piecewise continuous derivative on $[0,a]$, then $(f_N)$ converges uniformly to $f$ over $\Bbb R$. With: $...
0
votes
1answer
47 views

Convolution of Schwartz and test function approximated by partition of unity.

Let $\rho\in\mathscr{D}$, $0\leq\rho\leq 1, \rho(0) = 1$, and $\sum_{n\in\mathbb{Z}^d}$ $\rho(x-n) = 1$. Denote, $\rho_{n,\epsilon}() = \tau_n\rho(\frac{x}{\epsilon})$, where $\tau$ is the translation ...
2
votes
1answer
48 views

Coercitivity of an elliptic operator with constant coefficients

We are given an elliptic operator $P=\sum_{|\alpha|\leq m}a_\alpha\partial^\alpha$ that is elliptic in $\Omega$. $a_\alpha$ are constants. I am supposed to show that $$\|u\|_s\leq C_s(\|u\|_0+\|Pu\|_{...
2
votes
1answer
27 views

If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$.

If $\mu$ equals Haar measure on the 3-dimensional unit sphere $S^2$, then $\hat{\mu}(\varepsilon) = \dfrac{2\sin(2\pi |\varepsilon|)}{|\varepsilon|}$. I am not quite sure how to start this problem. ...
1
vote
0answers
27 views

How do I calculate the Fourier Transform of this signal?

The Context: Find $X(ω)$ which is the frequency domain representations of $x(t)$. $$ x(t) = \sum_{k=-\infty}^\infty \delta(t-4k) $$ This my professor's solution: As we can see, the ...
0
votes
1answer
14 views

Prove that if $f \in C^r(T)$, then $\hat{f}(n) = o(\frac{1}{|n|^r} )$ as $n \rightarrow ±∞$

I searched through everything that came up when I searched this question, but didn't come with anything. I'm used to typing in latex, so please excuse any formatting errors.
1
vote
0answers
27 views

Use trigonometric polynomial to approximate periodic function.

From page 53 of Fourier Analysis by Stein, we have If $f$ is integrable on the circle, then the Fourier series of $f$ is Cesaro summable to $f$ at every point of continuity of $f$. Moreover, ...
1
vote
0answers
27 views

Fourier series in spherical coordinates?

I'm reading an article and he just state: let $f\left(\theta,\varphi\right)$ be of this form $$f\left(\theta,\varphi\right)={\sum}g_{m}\left(\theta\right)e^{im\varphi},$$ I'm on the unitary ...
1
vote
0answers
70 views

Given that $\sum\frac{1}{n^2} = \frac{\pi^2}{6}$, how can I find $\sum\frac{1}{n^6}$?

Given that $\sum\frac{1}{n^2} = \frac{\pi^2}{6}$, how can I find $\sum\frac{1}{n^6}$? I know that you can prove the first equality using Fourier analysis. For the second one, do I try to use a ...
0
votes
1answer
35 views

Determing an inverse Fourier transform

The inverse Fourier transform is defined as: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$ I can't get an inverse Fourier Transform to Q1: $...
3
votes
1answer
32 views

What is duality argument for the operator on $L^p-$ spaces?

Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$ At various, places we see that (for ...
0
votes
1answer
16 views

inverse fourier transform of w*e^w

I have the function \begin{align} F^{-1}\{{λe^{-|λ|}}\} \end{align} How can we find the inverse Fourier transform? The correct answer is: \begin{align} \frac{-2ix}{π(1+x^2)^2} \end{align} Can ...
1
vote
0answers
33 views

Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of $\...
2
votes
1answer
67 views

Fourier series of dirac delta

Let $f \in S(\mathbb{R}^n)$ is it true that $$\frac{1}{(2\pi)^n} \lim_{\epsilon \rightarrow 0} \sum_{z\in \mathbb{}{Z^n}} \int_\mathbb{R^n} f\left( \frac{x}{\epsilon} \right) e^{iz (x-a\epsilon)} dx = ...
3
votes
0answers
34 views

Question about Dirichlet kernel of Fourier transform for $f\in L^p$ with $p\in [1,2]$, help needed in understanding proof.

I am trying to understand the proof that the following two statements are equivalent. For fixed $R>0$ and $f\in L^p(\mathbb{R}^n)$ let $$S_Rf(x)=\int_{|\xi|<R} \hat{f}(\xi)e^{2\pi i x . \xi}\,d\...
0
votes
1answer
16 views

Can we relax the hypothesis of Uniqueness theorem for Fourier series?

I know this fact: "Suppose that $f\in L^{1}(\mathbb T)$ and $\hat{f}(n)=0$ for all $n\in \mathbb Z,$ then $f=0 $ all most everywhere on $\mathbb T$." My Question is: Suppose that $f\in L^{1}(\...
1
vote
0answers
15 views

Sobolev space membership of logarithmic function

Determine the largest $s\in(0,1)$ for which the following integral converges $$\int_0^1\int_0^1\frac{\Big|\log|x-\frac{1}{2}|-\log|y-\frac{1}{2}|\Big|}{|x-y|^{1+2s}}^{2}dxdy$$
1
vote
0answers
22 views

Is $A(D)$ a complemented subspace of $C(T)$?

Let $T$ be the unit circle and $D$ the open unit disk. A function $f$ belongs to $C(T)$ if it is continuous at $T$. A function $g$ belongs to $A(D)$ if it is continuous at $\overline{D}$ and ...
0
votes
0answers
18 views

Fourier transform of integral with isotropic kernel

The textbook I'm reading claims that this integral: $$ A = \int_V \,d\mathbf{r} \int_V\,d\mathbf{r}' f(\mathbf{r}) K (| \mathbf{r} - \mathbf{r}'| ) f(\mathbf{r}')$$ can be written in Fourier ...
1
vote
1answer
51 views

Contradiction between Fourier and Laplace transforms?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has both Fourier and Laplace transforms. Also let $f(t)=0$ for all $t<0$. The Fourier transform of $f$ is $$\mathcal{F}(\omega)=\mathcal{F}...
1
vote
0answers
15 views

How can I convince myself of the Fourier scaling property via inverse FT?

I have this function $f(at)$, and I want to Fourier-tranform it. I proceed in the following way, for $\quad\alpha<0 \Longrightarrow a=-|a|$: \begin{align} \ \mathcal{F}_{t \rightarrow \xi}[f(at)]= ...
0
votes
0answers
19 views

What effect does sampling time have on a Fourier Series sum?

What effect would the sampling time of this Fourier sum have on it's accuracy? Is this to do with Nyquists theorem? or am I heading in the wrong direction with this question? Cheers
0
votes
1answer
33 views

Fourier transform of product of twho functions that includes characteristic function

I need to find the fourier transform of f(x)=(1-abs(x))(Chi_[-1,1] (x)). In words, I need the fourier transform of one minus the absolute value of x multiplied by the characteristic function on ...
0
votes
0answers
42 views

Sobolev and Fourier

If we have $f=1_{[a,b]} \varphi$ with $\varphi \in \mathcal{D}(\mathbb{R})$, we found that the sufficient and necessary conditions to have $f\in H^1$ is that $\varphi(a)= \varphi(b)=0$. If we take $\...
0
votes
0answers
33 views

Fourier sine and cosine series: reconstruction is shifted with respect to measured data

I am working in strain analysis. Strain in a mechanical testing machine is captured by strain gages. Signals are like the slim line in the graph below showing strain versus time. The data are of the ...
0
votes
0answers
29 views

How to solve $u_{tt}+\Delta u + x^{2}u=0$?

Let $u:\mathbb R \times \mathbb R \to \mathbb C$ be some function so the everything in the following make sense. Consider the following PDE: $\frac{\partial^{2}}{\partial t^2} u(x,t) + \frac{\...
0
votes
0answers
27 views

A $H^p$ function

Set $\mathbb U=\{x+iy|\;y>0\}$. A function $f:\mathbb U\to\mathbb C$ is called a $H^p$ function if $f(z)$ is holomorphic and $\|f\|_{H^p}:=\sup_{y>0} \left(\int_{-\infty}^{\infty} |f(x+iy)|^p dx\...
5
votes
1answer
49 views

What am I doing wrong when I try to deduce the Laplace transform formula?

The Laplace transform of a function $f(t)$ is the projection of $f(t)$ vector (indexed with $t$) onto the linearly independent set of vectors $e^{st}$. The projection of a vector $\vec{v}$ onto ...
1
vote
1answer
21 views

Relation between Fourier Transform Duality and other properties.

I'm having a hard time with Fourier Transform's Duality Property. The Duality Property states that, if $$\mathcal{F}\left\{x(t)\right\} = X(\nu),$$ then $$\mathcal{F}\left\{X(t)\right\} = 2\pi x(-\nu)...
1
vote
1answer
35 views

Asymptotic behavior of Fourier transform and Hölder condition

I'm trying to solve this question. Following the hint, the Fourier inversion formula gives me : $$ \big| f(x+h) - f(x)\big| = \left| \frac{1}{2\pi} \int_{-\infty}^\infty \widehat{f}(\xi) e^{i(x+h)\...