0
votes
0answers
66 views

Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
3
votes
1answer
44 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
2
votes
1answer
29 views

Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?

Let $C$ is a compact subset of $\mathbb R,$ $V\subset \mathbb R,$ and $0<m(V)<\infty,$ where $m$ is a Lebsgue measure on $\mathbb R.$ My Question is: Can we expect to find $k\in ...
0
votes
1answer
19 views

Pseudodifferential operators and amplitudes

I am studying psudodifferential operators on $\mathbb{R}^n$. Let $U\subset \mathbb{R}^n$ an open subset. A function is $b\in C^\infty(U\times U\times U \times \mathbb{R}^n)$ is an amplitude of order ...
2
votes
1answer
60 views

the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
2
votes
1answer
42 views

Sobolev spaces and Holder continuity (or, fractional derivatives and singularities)

I have two specific questions. The first is the result I actually need, and the second would let me prove it. EDIT: The second statement was wrong. I am keeping it for posterity. I am adding a third ...
6
votes
3answers
103 views

Eigenfunctions of the Laplace-Beltrami operator of a torus

The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known. What happens if we change the sides of the torus ...
0
votes
1answer
19 views

Fourier transforms similar $\Rightarrow$ functions similar?

I am wondering if there is a theorem that states something like the following? If $$\big|\;\tilde f(\omega)-\tilde g(\omega)\,\big| < \varepsilon\qquad \forall\omega$$ then there exists a ...
2
votes
0answers
35 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...
1
vote
1answer
32 views

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. ...
2
votes
1answer
23 views

Mean fourier coefficients of a $2\pi$-periodic function are just the usual Fourier coefficients.

Let $f$ be a continuous $2\pi$-periodic function on $\mathbb{R}$. I'm trying to show that \begin{align}\tag{1} \lim_{T \rightarrow \infty} \frac{1}{2T}\int_{-T}^{T} f(x)e^{-ix\xi}dx = ...
2
votes
0answers
34 views

Finding the sum of a trigonometric series, fourier series

I need to compute that for $x \in [0, 2\pi]$ $$\sum_{n=1}^\infty\frac{\sin(nx)}{n^3} = \frac{1}{12}x(x-\pi)(x-2\pi)$$ by using the uniform convergence $$\sum_{n=1}^\infty\frac{\sin(nx)}{n} = ...
1
vote
1answer
66 views

a generalization of normal distribution to the complex case: complex integral over the real line

How to prove $\int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi}$ for any $t\in \mathbb{R}$? I only obtained the case that $t=0$, $\int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}$. Thanks.
0
votes
0answers
20 views

compute the derivative of a function defined by an integration over the whole real line

Let $h(t)=\int _{\mathbb{R}}e^{-\frac{(x+it)^2}{2}}d\lambda(x)$, where $\lambda$ is the Lebesgue measure. I want to prove that $h$ is differentiable and compute the derivative of $h$: ...
2
votes
1answer
98 views

$\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$ [duplicate]

How we can do this sum? $$ \sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90} $$ I know that we could possibly use a Fourier series decomposition however I don't know what function to start with. I ...
0
votes
0answers
14 views

$ S_\lambda f = \sum_{j=1}^\infty \chi_{(-\infty,\lambda_j]}E_j f =(2\pi)^{-1}\int_{\mathbb{R}}\widehat{\chi}_{(-\infty,\lambda]} (t)e^{itP}f \,dt$

Let $P$ be a self-adjoint elliptic pseudo-differential operator on $M$ (compact manifold). I can not demonstrate that $$ S_\lambda f = \sum_{j=1}^\infty \chi_{(-\infty,\lambda_j]}E_j f ...
0
votes
0answers
22 views

Heisenberg uncertainty principle in D-dimensional

For Heisenberg uncertainty principle in D-dimensional there is $d^2$ in the formula.where does this additional term comes compared with the case of one dimensional?
2
votes
0answers
29 views

Good family of kernels in $\mathbb{R}^n$

I'm trying to prove that, given the heat equation $u_t = \Delta u$ with boundary values $u(x,0) = f(x)$, the solution given by $$u(x,t) = f \star H_t^{(d)}(x)$$ is continuous up to the boundary ...
1
vote
1answer
31 views

pointwise limit of uniformly bounded sequence in $A(\mathbb T)$ is again in $A(\mathbb T)$?

Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
1
vote
0answers
19 views

$f\mapsto \sum_{n\in \mathbb Z} |\widehat{F(f)}(n)|$ lower semi continuous?

Let $T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
3
votes
1answer
203 views

Integrate $\int_{-\infty}^{\infty}\exp\left(-\frac{\pi^2t(2x+1)^2}{2c^2}\right)\cos\left(\frac{(2x+1)\pi y}{c}\right)\exp(-2\pi i kx)dx$

By the poisson summation formula we have: $$\frac{1}{c}\sum\limits_{k=-\infty}^{\infty} \exp\left(-\frac{\pi^2t(2k+1)^2}{2c^2}\right)\cos\left(\frac{(2k+1)\pi ...
0
votes
1answer
38 views

How to interchange sum and integral when measure is in terms of Dirac measure?

Let $\{c_{k}\}_{k\in \mathbb Z} \subset \mathbb C$ such that, $\sum_{k\in \mathbb Z} |c_{k}| < \infty.$ Let $\delta_{k}$ is the unit Dirac mass at $k $, we note that $\mu = \sum_{k\in \mathbb Z} ...
0
votes
0answers
36 views

about lp estimate of Schwartz function

A homework question that I really couldn't find how to start. Prove that for any $f$ in Schwartz class $ \lVert f \rVert_{q} \leq C_{p,q} \lVert \nabla f \rVert_{2}^{a} \lVert f \rVert_{2}^{1-a} $, ...
0
votes
1answer
32 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
1
vote
0answers
19 views

Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
2
votes
0answers
38 views

Density of a set of functions in Schwartz space

I have a difficulty doing the following problem: Let $S(\mathbb{R}^n)$ be the Schwartz space. I need to determine whether the following set of functions $A$: $$A= \{f\in S(\mathbb{R}^n): ...
0
votes
0answers
20 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
0
votes
1answer
28 views

$f, \hat{f} \in L^{1}(\mathbb R) \implies \widehat {\text{Re}(f)}, \widehat{\text{Im}(f)} \in L^{1}(\mathbb R)$?

Let $f:\mathbb R \to \mathbb C$ such that $f(x)= (f_{1}(x), f_{2}(x))$; where $f_{1}(x)=\text{Re}(f(x))=\text{the real part of} \ f $ and $f_{2}(x)=\text{Im}(f(x))= \text{ the imaginary part of} \ ...
0
votes
1answer
37 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
2
votes
1answer
17 views

how to prove translation identity: $(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x)$?

Let $m, f\in S(\mathbb R^{n})$=The Schwartz space. My question: How to prove: $$(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x);$$ where ...
0
votes
0answers
23 views

properties of frequency- decomostion operator $\square_{k}^{\sigma}=\sum_{|\ell|_{\infty}\leq 1}\square_{k}^{\sigma}\square_{k+\ell}^{\phi}$?

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
0
votes
1answer
21 views

$\int_{|x|<t} |\mathcal{F}^{-1}f(x) |dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$?

Let $f\in L^{2}(\mathbb R^{n}).$ Fix $t>0,$ My Question:How to show, $\int_{|x|<t} |\mathcal{F}^{-1}f(x)| dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$ ? [We note $\mathcal{F}$ denotes the ...
1
vote
1answer
51 views

How to find the Fourier transform of $\mathbf 1_{[0,2\pi]}(x)\sin(x)$?

How does one find the Fourier transform of $f(x):=\mathbf 1_{[0,2\pi]}(x)\sin(x)$? I have tried to use the definition from my text: \begin{align*} \hat f(\xi) & = \frac{1}{\sqrt{2\pi}} ...
2
votes
1answer
51 views

Recover Fundamental solution of wave equation on $\mathbb{R}^n$ by on the sphere

It's well known that $\frac{\sin{t\sqrt{-\Delta}}}{\sqrt{-\Delta}}\delta$, the fundamental solution of wave equation on the $\mathbb{R}^n$ can be expressed as the form \begin{equation} \lim_{t\to ...
2
votes
0answers
41 views

Essential hypothesis of Fourier Inversion formula

Let $f\in L^{1}(\mathbb R)$ and we define its Fourier transform as follows: $\hat{f}(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i \xi\cdot x} dx, (\xi \in \mathbb R);$ and we define $f^{\vee}(x):=\hat{f}(-x)= ...
10
votes
4answers
607 views

Why would you expand a square wave in a Fourier series?

The periodic square wave $$ f(x) = \cases{ 1 \text{ if } 0 \le x \le \pi \\ 0 \text { if } -\pi \le x < 0} $$ seems easy enough to work with. Why transform it into a series of sines and cosines? ...
0
votes
1answer
10 views

Decay of fourier transform of a function with two variables

In the paragraph about 'use in harmonic analysis', the modulus of fourier transform of a (one variable) function $f(\xi)$ is shown to be bounded by a quantity involving powers of $\xi$. ...
1
vote
2answers
43 views

Fejer Kernel is Unbounded

Statement: Given the Fejer Kernel $F_n(x) = \frac{1}{n}\bigg(\frac{\sin(\frac{nx}{2})}{\sin(\frac{x}{2})}\bigg)^2$. Show that $F_n(x)$ is unbounded for $x=0$ as $n\rightarrow \infty$
1
vote
0answers
52 views

Uniqueness of Fourier Coefficients

I'm reading through Stein & Shakarchi's book on Fourier Analysis on my own, and have a question about the proof of the following theorem: Suppose that $f$ is an integrable function on the circle ...
1
vote
0answers
34 views

The property of positive fourier series. [duplicate]

This is the problem in the book 'Classical and multilinear harmonic analysis, volume 1' Let $f(x)=\sum_{n=0}^{N}[a_{n}\cos{2\pi nx}+b_{n}\sin{2\pi nx}]$ be a nonnegative function defiend on $[0,1]$. ...
0
votes
1answer
25 views

Why $\displaystyle\rho_{\alpha, \beta}(f)=\sup_{x\in\mathbb R^n}|x^\beta \partial^\alpha f(x)|$ is not a norm on $\mathcal{S}(\mathbb R^n)$?

The Schwartz space $\mathcal{S}(\mathbb R^n)$ is the set of all function $f:\mathbb R^n\longrightarrow \mathbb C$ such that $\displaystyle \sup_{x\in \mathbb R^n}|x^\beta \partial^\alpha ...
2
votes
0answers
22 views

Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$ in the Banach algebra $A(\mathbb R)$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
0
votes
1answer
25 views

What is $|x|$ for $x\in \mathbb T^n$?

The $n$-dimensional torus is $\mathbb T^n=\mathbb R^n/\mathbb Z^n$. Let $|x|$ be the Euclidian norm. What is $|x|$ for $x\in \mathbb T^n$?
0
votes
0answers
13 views

How to use $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in Banach sapace $C([0, T];M^{p,1})$?

(For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ...
0
votes
0answers
18 views

Find $g\in C^\infty(\mathbb R^n)$ one-periodic satisfying the identity below..

Let $f\in L^1(\mathbb R^n)$ and $k\in C^\infty(\mathbb R^n\times \mathbb R^n)$ be a $1$-periodic function in both variables. Is there a $1$-periodic function $g\in C^\infty(\mathbb R^n)$ satisfying: ...
3
votes
0answers
70 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
0
votes
1answer
37 views

A Specific Example about Parabolic PDE

I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one? The equation ...
0
votes
1answer
111 views

How to use Parseval' s( Plancherel' s) identity?

Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put, $F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt, \ (n=1,2,...).$ Fix $\alpha \in (0, \infty)$ and we define $H_{n}(x)$ ...
1
vote
0answers
37 views

How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
3
votes
0answers
53 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...