All Questions
13 questions
2
votes
0
answers
80
views
Prove uniqueness of Radon transform without using Fourier transform
The uniqueness of Radon transform can be expressed by the following claim (I assumed that the function has compact support for simplicity):
If a continuous function with compact support has zero ...
- 807
0
votes
1
answer
255
views
Carleson's theorem: proof of a lemma
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...
- 75
-1
votes
1
answer
213
views
Building a smooth function from a rapidly decreasing sequence
Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c_k\rbrace _{k \...
- 581
4
votes
1
answer
662
views
The decay of Fourier coefficients and the continuity of functions
Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not ...
- 1,219
1
vote
2
answers
152
views
Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?
Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
- 3,625
6
votes
1
answer
397
views
Absolute values of two functions and absolute values of their Fourier transform coincides
Let $f, g \in L^2(\mathbb{R})$.
Is it true that if both $|f|=|g|$ and $|\hat f|=|\hat g|$ hold, then there exists $\theta \in \mathbb{R}$ such that $f=ge^{i\theta}$?
I am not able to prove it or ...
- 489
2
votes
2
answers
333
views
Estimate for a simple oscillatory integral
If $\varphi$ is a smooth function on $\mathbb{R}$, then integration by parts implies that there exists a constant $C>0$ such that
$$
\Big|\int_0^1 \varphi(x)\, e^{i \lambda x}\, dx\Big|<\frac{C}\...
- 421
3
votes
1
answer
404
views
The sign of the tail of Fourier transform of a positive function/ characteristic function
I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c_\...
- 178
1
vote
0
answers
124
views
Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user89890
3
votes
0
answers
214
views
Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698
1
vote
1
answer
460
views
Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
I am reading a book "Fourier Series and Integrals" by Dym & McKean.
There is an exercise (Page 106):
Exercise: Check that if $f$ is a real, even, summable function and
if $f(0+)$ and $f(0-)$...
- 371
-1
votes
1
answer
1k
views
A question about pointwise convergence of Fourier transform in $N$-dimensions
I am retreating back on this statement, after some explorations and calculation
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
- 698
17
votes
2
answers
4k
views
Is this statement which relates the Fourier transform of a function to its singularities correct?
I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
- 698