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The log kernel and Bochner Theorem

I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that $$ L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t) $$ for every $x\in [0,1/2]$. On a structural ground, this ...
Adrien Hardy's user avatar
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5 votes
0 answers
194 views

When does the Fourier transform of a measure decay?

Let $\mu$ be a Borel measure on $\Bbb R^d$. It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies $$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$ However if ...
Guy Fsone's user avatar
  • 1,101
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
4 votes
0 answers
158 views

Measurability of $L^{p}(L^{q})$ integrable functions

Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that $ \int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty $ In addition we ...
User091819's user avatar
2 votes
0 answers
164 views

(Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis

It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
BigbearZzz's user avatar
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2 votes
0 answers
78 views

Definition of a continuous Gabor frame

I am trying to understand the definition of a Gabor frame and would appreciate some clarification with terminology. Let us begin with the setup: Let $G$ be a locally compact abelian group, and let $\...
Jake Wetlock's user avatar
  • 1,144
2 votes
0 answers
89 views

Prove integral inequality for divergence-free vector fields

Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold? $$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \...
Riku's user avatar
  • 839
1 vote
0 answers
105 views

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
mathlover's user avatar
1 vote
1 answer
294 views

Regarding subspace generated by the polynomial multiples of outer functions

Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
user429197's user avatar
1 vote
0 answers
129 views

Differentiability of $f*g$ on the circle, for integrable f, bounded g, and some decay of the Fourier coefficients of f

If $f\in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$ where $\mathbb{T}$ is the circle, such that $\hat{f}\in L^{p}(\mathbb{Z})$ for some $1\leq p<\infty$, do we have that $f*g$ is ...
Alin Galatan's user avatar
0 votes
0 answers
75 views

Extracting the point mass measure of some type of positive measures

Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals. Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
ABB's user avatar
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