All Questions
30 questions
5
votes
0
answers
160
views
Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
- 929
1
vote
1
answer
151
views
Some operators on spheres
Let $S_2$ be the unit sphere in $\mathbb R^3$ equipped with normalized Haar measure. For a continuous function f and $\delta\in (-1,1)$ define $T_\delta f(x):=\int_{\{y:<x,y>=\delta\}}f(y)d_\...
- 1,879
0
votes
1
answer
123
views
Proving a Fourier transform inequality for functions with mixed variable bounded support
I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide.
Let $\gamma\...
- 131
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 $A$ ...
- 29
1
vote
1
answer
133
views
A question about the maximal function
Let $n>4$, $f\in C^{\infty}(\mathbb{R}^{n})$ and 0 denote the origin of $\mathbb{R}^{n}$. We define a weighted maximal function by $$Mf(x)=\sup_{0<r<1}r^{4-n}\int_{B_{r}(x)}|f|$$ which is ...
- 155
1
vote
1
answer
137
views
Can functions with "big" discontinuities be in $H^1$?
How can I prove that the function:
$$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
- 1,759
0
votes
1
answer
161
views
Does convolution with $(1+|x|)^{-n}$ define an operator $L^p(\mathbb R^n) \to L^p(\mathbb R^n)$
Suppose that $f : \mathbb R^n \to \mathbb R^n$ is a locally integrable function. I am interested in the integral
$$ x \to \int_{\mathbb R^n} ( 1 + |y| )^{-n} f(x-y) \;dy $$
If the decay of the ...
- 641
4
votes
0
answers
495
views
Fixing the duality $L^\infty(X)= L^1(X)^*$ for Radon measure spaces
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
Let me denote the Borel subsets of $X$ by $\mathscr{B}(X)$. Folland claims that if $\mu$ is a ...
- 175
2
votes
1
answer
261
views
Qualitative difference between "continuous" and "discontinuous" states on $M(G)$
Let $G$ be a locally compact Abelian group (we can think that $G={\mathbb R}$). Let $C_0(G)$ be the space of continuous functions $u:G\to{\mathbb C}$ vanishing at infinity with the usual $\sup$-norm, ...
- 7,426
1
vote
0
answers
64
views
Characterization of elements of Hardy Space
Let $\Omega\subset\mathbb{C}^n$ be a $C^{\infty}$ bounded domain. Let $H^2(\partial\Omega)$ denote the Hardy space, and $S(.,.)$ denote its Szego Kernel. We know that
$$
\forall f\in H^2(\partial\...
- 63
2
votes
1
answer
128
views
Regarding basis of holomorphic Hardy space
Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain and let $H^2(\partial\Omega)$ denotes a holomorphic Hardy space which is a $L^2(\partial\Omega)$ closure of $A^{\infty}(\Omega)(=\...
- 63
5
votes
1
answer
299
views
About weak integrals: Appendix of Folland's book "A course in abstract harmonic analysis"
Consider the following fragment from Folland's book "A course in abstract harmonic analysis":
All integrals are here to interpreted in the weak sense (see p285 in Folland's book). Why is ...
- 175
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 ...
- 4,058
0
votes
1
answer
88
views
An equation in the convolution measure algebra on reals
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals.
Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta_0$ be the point mass measure concentrated on ...
- 4,058
5
votes
1
answer
495
views
Unifying two definitions of $L^\infty$
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$.
Definitions:
A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ ...
- 175
2
votes
1
answer
223
views
Show that $V_G: L^2(G\times G, \mu \times \mu)\to L^2(G\times G, \mu \times \mu)$ defined by $V_G(f)(x,y) = f(xy,y)$ is well-defined
Let $G$ be a locally compact Hausdorff group and let $\mu$ be a right Haar measure on $G$. Then $\mu\times \mu$ (the Radon product of measures) is a right Haar measure on $G \times G$ and we can ...
- 175
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 ...
- 1,245
7
votes
1
answer
1k
views
Properties of convolutions
Consider the function
$$f_{n}(x)=e^{-x^2}x^n.$$
and the function
$$h_p(x):=e^{-\vert x \vert^p}.$$
My goal is to analyze
$$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
- 173
1
vote
2
answers
226
views
Number theory on Banach space $L^2(\mathbb R)$ meets linear independence?
Consider an orthonormal basis $(\varphi_k)$ of $L^2(\mathbb R)$ with Lebesgue measure.
I came along a nice number theoretic question in analysis:
Write $$f_k(x):=\int_{\left\lvert y \right\rvert \...
- 25
1
vote
1
answer
52
views
Infinitely many independent functions that are only frequency localized?
A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
- 11
2
votes
1
answer
311
views
Differentiation on $[0,1]$
EDIT:
Perhaps a more reasonable question after thinking about the answer I got would have been.
Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
- 536
2
votes
1
answer
1k
views
Pointwise convergence implies uniform convergence?
Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like
$$(Sf)(x)= \int_{\mathbb{R}^n}K(x,y)f(y)dy.$$
Assume that $K\in C^{\text{bounded}...
- 329
4
votes
1
answer
495
views
Weil's Haar measure construction from below
Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function.
I would need to know something similar for an ...
user1688
0
votes
1
answer
193
views
$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$
Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
2
votes
1
answer
313
views
Spherical decreasing rearrangement on the sphere
On $\mathbb{R}^n$, we have the concept of spherically decreasing rearragement of a function, which means, given a function $f$, one can design a radial and decreasing function $f^*$ such that $\Vert f^...
- 21
0
votes
0
answers
405
views
What does the Plancherel theorem say about positive-definite distributions?
I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem.
The ...
- 3,799
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 ...
- 1,051
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 ...
- 647
1
vote
0
answers
154
views
A problem concerning measures on locally compact spaces
I am stuck on a question for quite sometime now, although in the text it is said to be "apparent". The problem goes as the following :
Let $X$ and $Y$ be locally compact Hausdorff spaces. Then $M(X)$ ...
- 111
0
votes
1
answer
720
views
Pointwise limit at Lebesgue's point
Dear MOs,
I am sorry if this problem is too elementary for someone. I just want to get confirmation.
Suppose $f\in L^1(R^d)$. Since almost all points are Lebesgue points by the Lebesgue ...
- 1,649