4
votes
2answers
98 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} ...
0
votes
1answer
142 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$? [closed]

(This may be very simple question for MO; I had post it to math stack exchange few days back but I could not get any answer(or comment) to it) It is well-known that, for $f,g \in L^{1}(\mathbb R).$ ...
18
votes
2answers
2k 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 ...
3
votes
0answers
109 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 ...
0
votes
1answer
167 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 ...
-3
votes
1answer
162 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 ...
1
vote
1answer
133 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 ...
1
vote
1answer
259 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 ...
8
votes
0answers
198 views

Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the Fourier transform and decompose $L^1$ ...
0
votes
1answer
121 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 ...
0
votes
0answers
56 views

Factorization in Fourier Algebra and its properties

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, ($\ast$ stands for a usual convolution ) $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): ...
1
vote
0answers
115 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 ...
12
votes
1answer
1k 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 ...
0
votes
1answer
250 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 ...
17
votes
2answers
607 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 ...