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Composition of Riesz potentials

For $0<\alpha<n$ and $n\geq 2$ we define the Riesz potential by $$ (I_\alpha f)(x) = \frac{1}{\gamma(\alpha)} \int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}}\, dy\, , \quad \text{where} \quad \...
Piotr Hajlasz's user avatar
11 votes
1 answer
691 views

Reference request: Fourier transform on the multiplicative group of real numbers

Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with ...
Jochen Glueck's user avatar
2 votes
0 answers
143 views

Need to show bounded behavior of a particular Fourier transform

First let me be briefly state the relevant information to my problem: $\beta(s) \in C_0^{\infty}([-1,1])$, and $\beta \equiv 1$ around $s=0$. The $\beta$ I'm using is an even function, but it doesn't ...
Patch's user avatar
  • 377
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
1 vote
1 answer
180 views

Annihilator property dual

Let $G$ be a locally compact group and $\phi$ be in $ L^{\infty}(G)$ that annihilates $I$, where $I$ is a closed ideal of $ L^1(G)$, so by duality we have: $$\int_G f(y)\phi(y)dy=0$$ for all $f\in I$....
M.fouladi's user avatar
  • 399
6 votes
0 answers
203 views

Uniform estimates of Fourier transform of tempered functions with parameters

Consider the following function in $\mathbb{R}^3$: $$ f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2}, $$ where $\frac{1}{2}<\alpha<1$, and ...
Tomas's user avatar
  • 879
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
2 votes
1 answer
190 views

Half Poisson summation

Suppose $f$ is a Schwartz function on $\mathbb{R}$. Is there a closed formula for $$\sum_0^\infty \hat{f}(n)$$ where $\hat{f}$ is the $n$-th Fourier coefficient of $f$?
Qijun Tan's user avatar
  • 587
3 votes
0 answers
651 views

Decay of the Fourier transform of a surface area measure

Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...
Alan Watts's user avatar
1 vote
0 answers
327 views

If $\mathcal{F}$ is the Fourier transform, what can be said about $\mathcal{F}(L^1(\mathbb{R})) \cap L^1(\mathbb{R})$?

The Fourier transform gives a map of the Schwartz space to itself which turns out to be a linear homeomorphism of period 4. However, when the domain is extended to $L^1(\mathbb{R})$, the situation is ...
juan arroyo's user avatar
3 votes
2 answers
1k views

Are the zeroes of the Fourier Transform of compactly supported functions isolated?

I have a continuous function $f$ on a locally compact Abelian group $G$ with compact support, and I would like to say that the zeroes of $f$ are sparse in some sense (isolated would be good, uniformly ...
Nick S's user avatar
  • 2,071
2 votes
0 answers
215 views

Generalization of Pitt's theorem

Pitt's theorem (Pitt 1937), states that the one-dimensional Fourier tranform is well defined and continuous between the weighted spaces $L^p(\mathbb{R},|x|^{bp}dx)$ and $L^q(\mathbb{R},|x|^{\beta q}dx)...
Capublanca's user avatar
1 vote
1 answer
359 views

Relating the R-transform in free probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
pre-kidney's user avatar
  • 1,329
1 vote
0 answers
157 views

Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$ or to know where I can find some techniques to ...
Felice Iandoli's user avatar
0 votes
0 answers
79 views

Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?

I have been learning Wiener amalgam spaces. In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$ Take $f(x)= \chi_{\...
Inquisitive's user avatar
  • 1,051
10 votes
2 answers
6k views

Fourier transform of compactly supported distribution is smooth

My advisor made the comment that if $u\in \mathcal{E}'$ is a compactly supported distribution, then $\hat{u}(\xi)\in C^{\infty}(\mathbb{R}^n)$ is actually a smooth function (not merely a distribution ...
Patch's user avatar
  • 377
3 votes
1 answer
518 views

Connection between the Fourier transform of f and |f|

If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and $$ \|\widehat{f}\|_{L^{p'}}\...
Wang Ming's user avatar
  • 425
4 votes
3 answers
334 views

Is there a compactly supported function that its Fourier transfrom vanishes at given n real points?

My question is as follows: Given ${{\lambda }_{1}},\,{{\lambda }_{2}},...,{{\lambda }_{n}}\in \mathbb{R}$ where $\underset{1\le j\le n-1}{\mathop{\min }}\,\left| {{\lambda }_{j+1}}-{{\lambda }_{j}} \...
Baily's user avatar
  • 141
2 votes
0 answers
120 views

request for any expository works in pointwise convergence of double Fourier series and especially a paper by Hardy

Quart. J. Math. Volume 37, Issue 1, Pages 53-79 On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Hardy, G.H. I am not ...
Rajesh D's user avatar
  • 698
-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
3 votes
2 answers
196 views

Inverse Fourier of $\omega^{-1+{\rm i}\alpha} u(\omega-1)$

Let $\alpha$ be an arbitrary real number and define \begin{align} \widehat{f}(\omega)=\left\{\begin{array}{ll} \omega^{-1+{\rm i}\alpha}, & \omega>1,\\ 0, & \textrm{otherwise}. \end{array} \...
Arash's user avatar
  • 31
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
5 votes
0 answers
286 views

$f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$ Let $1\leq p \leq ...
Inquisitive's user avatar
  • 1,051
1 vote
1 answer
289 views

Closed sets in the space of Fourier transforms $\mathcal{F}L^{1}$

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
Inquisitive's user avatar
  • 1,051
-3 votes
1 answer
230 views

$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at $\...
Inquisitive's user avatar
  • 1,051
1 vote
1 answer
229 views

Result of Beurling concerning absolute convergence of Fourier series of |f|

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put, $$A(\mathbb T):= \{f\in L^{1}(\...
Inquisitive's user avatar
  • 1,051
1 vote
1 answer
484 views

When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
Inquisitive's user avatar
  • 1,051
23 votes
0 answers
1k views

Laplace Transform in the context of Gelfand/Pontryagin

Questions: Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform? If not, is there a ...
Greg Zitelli's user avatar
  • 1,124
1 vote
1 answer
367 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)? [closed]

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
Inquisitive's user avatar
  • 1,051
1 vote
0 answers
181 views

How Fourier transform behaves if we kills the oscillation?

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put $$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$ and $$FL^{1}_{b}(...
Inquisitive's user avatar
  • 1,051
25 votes
3 answers
13k views

Fourier transform of the unit sphere

The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
Francois Ziegler's user avatar
0 votes
1 answer
489 views

Littlewood-Paley theory and norm estimation

In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1. It is claimed that Lemma 2 is ...
MichaelNgelo's user avatar
26 votes
2 answers
3k views

Image of L^1 under the Fourier Transform

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...
Francis Adams's user avatar

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