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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 ...
Zhang Yuhan's user avatar
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 ...
Alexander's user avatar
-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 \...
Peg Leg Jonathan's user avatar
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 ...
Luis Yanka Annalisc's user avatar
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{...
Johnny T.'s user avatar
  • 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 ...
J.Mayol's user avatar
  • 489
2 votes
2 answers
332 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}\...
Tony419's user avatar
  • 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_\...
Tanya Vladi's user avatar
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 ...
user avatar
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\...
Rajesh D's user avatar
  • 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 $f(0-)$...
Hheepp's user avatar
  • 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 ...
Rajesh D's user avatar
  • 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 ...
Rajesh D's user avatar
  • 698