2
votes
1answer
139 views
+50
Fourier analysis questions
Can anyone give me a hand with the proof of this properties?
Prove that:
a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$. ...
1
vote
3answers
37 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 ...
3
votes
1answer
40 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 ...
0
votes
0answers
22 views
DFT in complex form and Trigonometric Polynomial Interpolation: why different dimensions of basis vectors set?
I'm trying to figure out, how coefficients of Discrete Fourier Transform in complex form are converted into coefficients of Trigonometric Polynomials Interpolation:
Say, I have a function vector with ...
1
vote
1answer
24 views
Extracting Harmonic series components
I have a number which is made up of a Harmonic series.
1/2 + 1/3 + 1/4 etc.
Some of the components may not be in the number..
1/2 + 1/7 + 1/11 etc.
Is it possible to recover the individual ...
0
votes
0answers
16 views
Limit of integrals of a sequence of locally integrable continuous functions.
Suppose that $\sigma$ is locally integrable and continuous on $(\mathbf{R}^n)^m$. Fix $\vec a \in (\mathbf{R}^n)^m$ and let $f_1,\ldots,f_m$ be Schwartz functions on $\mathbf{R}^n$. In the proof I'm ...
1
vote
1answer
35 views
Showing a Schwartz Function Bound
I have a question on how to get the correct upper bound of a Schwartz function. Unfortunately, I've never understood this even though I've seen my professors do it a thousand times. I figured it's ...
3
votes
2answers
74 views
Identity involving partial sums of Fourier series
Suppose $f$ is a continuous periodic function and $S_Nf(x) = \sum^N_{n=−N} \hat f(n) e^{inx}$, where $$\hat f(n)= \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx} dx.$$
How can I show that ...
2
votes
1answer
23 views
Given $\widehat{\varphi} = \chi_{[-1/2,1/2]}$, show the system $\{\varphi(2^v x)\}_{v \in \mathbb{Z}}$ is not orthogonal
If $$\widehat{\varphi} = \chi_{[-\frac{1}{2},\frac{1}{2}]}$$ then it is not hard to compute via the inverse Fourier transform that $$\varphi(x) = \frac{\sin(\pi x)}{\pi x}$$ so we need to show ...
1
vote
0answers
73 views
Another aspect of Heisenberg uncertainty principle
In fourier transformation theory, we have the Heisenberg uncertainty principle, i.e.
Suppose $\phi$ is a function in [Schwarz space][1] which satisfies the normalizing condition ...
1
vote
1answer
39 views
$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq \int_{-\infty}^{\infty}|f|^2 dx$?
Suppose complex function $f$ in the Schwartz Space, its definition see http://en.wikipedia.org/wiki/Schwartz_space how can we argue that
$$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq ...
1
vote
1answer
71 views
Fourier transform of a function is square integrable
Is there a result stating that if a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is square integrable and decays at infinity, then its Fourier transform is also square integrable?
1
vote
0answers
24 views
$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$
Here $i$ is complex number, $n$ is positive integer. Show that
$$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$
This question appears from Stein's ...
1
vote
0answers
28 views
Variation on a completeness relation
The completeness relation for the spherical harmonics is:
$$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}^*\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) = ...
1
vote
1answer
49 views
Help proving Calderón reproducing formula (simple version)
Let $\phi$ be a real compactly supported smooth function on $\mathbb R$ with total integral zero. Define $\phi_t=\frac{1}{t} \phi(\frac{x}{t})$. I also suspect that they must be even, but the notes I ...
0
votes
0answers
37 views
Strichartz estimates and operator from $L^{2}_{x}$ to $L^{6}_{x,t}$
I want to prove that the operator $T=| \nabla|^{1/6} e^{-t\partial ^{3}_{x}}\tilde{P}_{N}$ takes functions from $L^{2}_{x}$ to $L^{6}_{x,t}$. The hint is to first prove for Schwartz functions, and ...
2
votes
1answer
74 views
Decaying Fourier transform and smoothness
Suppose that $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies
$$
|\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}.
$$
I want to show ...
3
votes
1answer
47 views
Continuous, integrable fourier transform of an $L^{2}(\mathbb{R})$ function, integrable.
I've come across a number of sources claiming a smoothness-decay duality between a function and its Fourier transform. But most seem to give results about how the smoothness of a function leads to ...
4
votes
1answer
120 views
Convolution square root of $\delta $
I want to somehow classify the distributional solutions of the equation
$$
f \ast f = \delta
$$
where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
3
votes
1answer
97 views
Dirichlet Problem: Uniqueness of solution
Let $u$ be the solution to a Dirichlet Problem on a bounded open domain $D \subset \Bbb R^n$.
Is the uniqueness of $u$ guaranteed by the maximum principle or by the smoothness of the boundary of $D$?
...
1
vote
1answer
53 views
Dirichlet problem: Obtaining the harmonic measure through Riesz representation theorem
For the Dirichlet problem on a bounded open domain $D \subset \Bbb R^n$
$$
\Delta u=0, \text{ on } D, \\
\left. u\right|_{\partial D}=f \in C\left( \partial D\right).
$$
With a fix $x$ in $D$, an ...
0
votes
0answers
83 views
Dirichlet Problem: Example where the Green function is not the Poisson kernel
Give an example of a Dirichlet problem where the Green function is not the Poisson kernel.
For a bounded open domain $D$ with a sufficiently smooth boundary and $f \in C\left(\partial D \right)$, the ...
1
vote
1answer
79 views
Dirichlet problem: Is the Poisson Integral always a solution?
Let $f$ be continuous on the sufficiently smooth boundary $\partial D$ of a domain $D \subset \Bbb R^n$.
Is the Poisson integral of $f$,
$$
Pf(x)=\int_{\partial D} f(t) ...
1
vote
0answers
119 views
Find a sequence of function in the Schwartz space $S(\mathbb R)$ which does not converge in $S(\mathbb R)$
Show there exists a sequence $\{f_n\}$ in the Schwartz space $S(\mathbb R)$ with limit $f$ for which
$$
\lim \|f_n\|_{u,v} \text{ induced that } f \not\in S(\mathbb R) \text{ for some } u,v.
$$
But
...
1
vote
0answers
55 views
Suggestion for a project on Harmonic measure and Fourier analysis
I have a course project on harmonic measure and Fourier analysis.
The goal is to give a presentation on a part of harmonic measure theory which relates to Fourier analysis.
Harmonic measure is a vast ...
2
votes
1answer
71 views
Proof for Fourier transform in $L^2$
This question makes me really confused:
Let $f$ and $g$ two functions in $L^2$. Show that:
$$\int \widehat f\cdot gdx= \int f\cdot\widehat gdx,$$
where $\widehat f$ is the Fourier transform ...
6
votes
0answers
193 views
Plancherel formula for compact groups from Peter-Weyl Theorem
I'm trying to derive the following Plancherel formula:
$$\|f\|^{2}=\sum_{\xi\in\widehat{G}}{\dim(V_{\xi})\|\widehat{f}(\xi)\|^{2}}$$
from the statement of the Peter-Weyl Theorem as given by Terence ...
2
votes
0answers
67 views
How is study of fractals related to fourier/spectral/harmonic analysis?
In chap. 3 of "Fractal Geometry of Nature" Mandelbrot mentions that "part of the study of fractals is the geometric face of harmonic analysis" (spectral or Fourier, he specifies). But to my dismay ...
0
votes
1answer
40 views
how to prove the convolution formular?
let $\overset{\backsim} {g}(x)=g(-x)$;
suppose $u,\phi,\psi$ always make the integral significant,$E_n$ is the n-dimensional euclidean space. Then how to prove
...
1
vote
1answer
97 views
If $f$ is a bounded tempered distribution and $g \in L^1$ is then $\int_{\Bbb R^n}(f\ast\tilde\varphi)(x)\tilde g(x)\,dx$ a tempered distribution?
Let $f$ be a bounded tempered distribution, that is, $f\ast\varphi \in L^\infty(\mathbb R^n) $ for every Schwartz function $\varphi$. If $g \in L^1(\mathbb R^n)$, does the following definition define ...
4
votes
0answers
59 views
multipliers on $H^{1}$
I'm begining to study the hardy space $H^{p}(\mathbb{R}^{n})$. First recall that a $L^{\infty}$ function is called a $H^{1}$ multiplier if the associated operator ...
3
votes
1answer
145 views
Bounds on integral
I am calculating Fourier coefficients for certain functions and have come across an integral of the form
$$I=\int_0^{2\pi} \int_0^1 r^2e^{2\pi i r(m\cos\theta+n\sin\theta)}drd\theta,$$
where ...
3
votes
0answers
45 views
$\Lambda_p$-set for compact abelian group
We denote by $|A|$ the cardinal of a set $A$. Let $S$ be a subset of $\mathbb{Z}$. Denote $S_N=S\cap [0,N]$ where $N$ is an integer. Suppose $2<p<\infty$. There is well-known that if $S$ is a ...
0
votes
0answers
31 views
boundedness of the Hilbert transform on Bochner spaces
Let $X$ be a Banach space. Let $H$ be the Hilbert transform (on $\mathbb{R}$ or $\mathbb{T}$). Suppose $1<p,q<\infty$. It is well-known that if $H \otimes Id_X$ is bounded on the Bochner space ...
2
votes
1answer
79 views
Problems on Schwartz Functions
(1) What are all positive Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ?
(2) What are all Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ?
(3) What ...
0
votes
1answer
94 views
Question from Stein's Singular Integrals and Differentiability Properties of Functions.
My question is in regards of Stein's proof that Hilbert transform is of weak $(1,1)$ property, on page 30 of the textbook I mentioned in my title.
On page 32 he writes that because $|\nabla K| \leq B ...
1
vote
0answers
142 views
A function in BMO space
Let $\psi:[0;1]\to\mathbb R$ is a nonnegative measurable function. Let $b_d(x)=1_{B(0,1)}\cdot{\rm sgn}(\sin (\pi d|x|))$, where $d\in\mathbb N$. Here $1_{B(0,1)}$ is the charateristic function of the ...
3
votes
2answers
208 views
Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space
Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...
3
votes
2answers
71 views
The integral of a function over $S^1$
Let $S^1=\mathbb R/\mathbb Z,$ I was wondering how to calculate the integral of a function over $S^1$ and why. Like, $\int_{S^1}1 dx=?$ Given an "appropriate" function $f$, what is $\int_{S^1}f(x)dx?$
...
1
vote
1answer
139 views
On covering lemma and Calderón–Zygmund decomposition
I am working on something which needs to understand covering lemmas and Calderón–Zygmund decomposition. These type of lemmas are as in the following link
...
2
votes
1answer
197 views
Applications of Young's convolution inequality
Recall that the convolution of two functions is given by
$$f*g(y)=\int f(x)g(y-x)dx.$$
The well known inequality known as Young's inequality, say that
$$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$
provided ...
1
vote
1answer
125 views
On a duality Fefferman-Stein's inequality
Let $M$ be the Hardy-Littlewood maximal operator. In the book "Weighted norm inequalities and Related Topics" by Rubio de Francia and J. Cuerva, page 150, theorem 2.1.2 states as the following:
*For ...
4
votes
1answer
131 views
Product of Sidon sets
Let $G$ be a compact abelian group with dual $\Gamma$. Let $\Lambda \subset \Gamma$ a Sidon set (see the book of Rudin: Fourier Analysis on Groups for the definition).
Consider the set ...
2
votes
1answer
184 views
Asymptotic error of Fourier series partial sum of sawtooth function
In Iwaniec's book, Topics in Classical Automorphic Forms, pg. 4, he gives the statement:
$$\{x\}=\frac{1}{2}-\sum_{n=1}^N\frac{\sin 2\pi nx}{\pi n}+O((1+||x||N)^{-1})$$
where $\{x\}$ denotes the ...
4
votes
1answer
2k views
Criteria for swapping integration and summation order
I have a function (a potential from an electrostatic potential via a Fourier series) in the form of
$$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m)
du\, dv$$
...
7
votes
1answer
172 views
Fourier transform of function in $L^{4/3}$
Suppose $f \in L^{4/3}(\mathbb{R}^2)$ and denote its Fourier transform by $\mathscr{F}(f)$. Is it true that the function $g:\mathbb{R}^2 \rightarrow \mathbb{C}$ defined by
...
1
vote
0answers
121 views
Estimate the Hilbert transform
Let $1\leq p<∞$: Suppose that there exists a constant $C>0$ such that for all $f\in S(\mathbb{R})$ with $L^p$ norm one we have $$\biggl|\{x:|H(f)(x)|>1\}\biggr|\leq C.$$ Here $H(f)$ is ...
1
vote
3answers
157 views
Why is it useful to express PDE solutions as $L^2$-convergent series?
The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...
3
votes
2answers
320 views
Why are translation invariant operators on $L^2$ multiplier operators
For $m \in L^\infty$, we can define the multiplier operator $T_m \in L(L^2,L^2)$ implicitly by
$\mathcal F (T_m f)(\xi) = m(\xi) \cdot (\mathcal F T_m)(\xi)$
where $\mathcal F$ is the Fourier ...
2
votes
1answer
167 views
Convolution inequality
Let $u$ and $v$ be two $L^1(\mathbb{R})$ functions such that
$\|u\|_{L^1} \le \|v\|_{L^1}$ and $f$ is non-negative $L^1(\mathbb{R})$
with non-negative inverse Fourier transform.
Is it true that for ...
