All Questions
Tagged with fa.functional-analysis harmonic-analysis
451 questions
23
votes
9
answers
2k
views
Nonseparable counterexamples in analysis
When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...
- 4,679
0
votes
1
answer
212
views
The Quotient exponential operator
I have a question if you don't mind. I have the following quotient operator:
$$\frac{1}{e^{d/dx}(f(x))}$$
Where $f$ is a smooth function on $R$. I would like to get rid of the denominator. IS there ...
- 6,696
4
votes
0
answers
149
views
Cyclic vectors for the translation operator
Let $b\in \mathbb{R}\neq 0$, and consider the translation operators:
$$
\begin{align}
T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\
f &\mapsto f(\cdot + b).
\end{align}
$$
*Are there known ...
- 5,405
2
votes
0
answers
191
views
Convergence in $S'(\mathbb R^d)$ of the paraproduct $\dot{T}_uv$
Let $B = B(0,4/3)$, $C = \{x \in \mathbb R^d : 3/4 \leq \|x\|_2 \leq 8/3\}$ and $\tilde{C} = \{x \in \mathbb R^d : 1/12 \leq \|x\|_2 \leq 10/3\}$.
For a fixed Littlewood-Paley decomposition $\chi \in \...
- 233
5
votes
0
answers
118
views
Good (Sidon) Approximation of "Bumps"
Given a rational point $p\in S^1$ and a continuous function $f:S^1\rightarrow \mathbb C$, we say that $f$ is an $\epsilon$-bump around $p$ (for some $\epsilon>0$) if $f(p)=1,|f|_{\infty}\leq 1+\...
- 113
3
votes
1
answer
191
views
A convolution type singular integral operator with log
Define a convolution type operator $T_m$ by
$$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer.
Consider $f \in H^s (s > 0)$ which is the usual Sobolev space. ...
- 903
3
votes
2
answers
203
views
What is the distribution of the following limit?
Assume $x \in \mathbb{R}$. We already know that
$$\lim_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta_x.$$
Here $\delta_x$ denotes the Dirac distribution. If we ...
- 266
4
votes
1
answer
2k
views
Norm of convolution operator
By Young's inequality for any $f\in L^p(\mathbf{R})$ the map $T_f:g\mapsto f\star g$ is a continuous operator from $L^q(\mathbf{R})$ to $L^r(\mathbf{R})$ where $1\leq p,q,r\leq \infty$ satisfy $1+\...
CommunityBot
- 1
1
vote
1
answer
279
views
Understanding the regular representation of an LCA group as a 'direct integral'
The reference for what I'm asking is page $107$ from Folland's harmonic analysis.
$G$ is a locally compact abelian group with dual $\hat{G}$. Let $H$ denote the Hilbert space $L^2(G)$.
I'm trying to ...
- 675
0
votes
1
answer
226
views
Transformation of Fourier Transform
Suppose that $f$ is a function with a Fourier transform, and that $g:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth function such that $g\circ f$ has a Fourier transform also.
Is there an expression ...
- 16.2k
6
votes
1
answer
365
views
Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra?
Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B_{p,q}^s(\mathbb{R}^d)$ defined by the norm
$$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$
...
- 39k
4
votes
1
answer
257
views
condition for $f \in H^{1/2} \cap C^0$
I need to show that for $f \in H^{1/2}(S^1) \cap C^0(S^1)$ we have :
$$ \iint\limits_{S^1 \times S^1}{ \frac{|f(x)-f(y)|^2}{\sin^2(\pi(x-y))}dx dy}< + \infty $$
and that, conversely, if $f$ is a ...
- 3,065
2
votes
2
answers
836
views
Laplace equation on the disk with Robin boundary condition
Consider the following two dimensional Laplace equation on the unit disk $D$ with homogeneous Robin boundary condition:
$$\Delta u = 0, ~~\frac{\partial u}{\partial n} = b(x) u(x)~~ \forall x \in \...
- 2,494
5
votes
2
answers
775
views
On inverting characteristic functions
Let $X$ be a random variable in $\mathbb{R}^n$ with distribution $\mu$ and characteristic function $\varphi$ (i.e. $\varphi(t)=\mathbb{E} e^{i\langle t,X\rangle}$). The standard inversion formula ...
- 8,992
25
votes
3
answers
13k
views
Fourier transform of the unit sphere
The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
- 14k
6
votes
1
answer
378
views
Wiener Corollary in "An introduction to harmonic analysis" by Yitzhak Katznelson
I can't understand a lemma in "An introduction to harmonic analysis" by Yitzhak Katznelson which is stated as follows:
Corollary. Let $\mu\in M(\mathbb T)$. Then
$$\sum\limits_{\tau\in\...
- 6,367
7
votes
2
answers
304
views
Existence of $f \in L^2(\Bbb R^n)$ with $f=g_1$ on $E$ and $\mathscr{F}(f)=g_2$ on $F$
The question has been posted here but had no response.
Question:
Suppose $E,F$ subsets of $\Bbb R^n$ have finite measure. Show that for any $g_1,g_2 \in L^2(\Bbb R^n)$ there exists $f \in L^2(\Bbb R^...
- 3,065
7
votes
0
answers
132
views
Smoothing property of a certain singular integral operator of non-convolution type
For simplicity, suppose that the dimension $d=2$, and let $g_s(x)$ be the Coulomb or Riesz potential defined by
$$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
- 873
3
votes
1
answer
203
views
Using Fourier series to prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$
Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)...
- 188k
2
votes
0
answers
169
views
Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
- 839
4
votes
2
answers
1k
views
Hardy-Littlewood-Sobolev inequality in Lorentz spaces
Hardy-Littlewood-Sobolev inequality states that if $1<p<q<\infty$, $1/r=1-1/p+1/q$, then we have
$$\left\|\frac{1}{|x|^{n/r}}\ast f\right\|_{L^q(\mathbb R^n)}\le\|f\|_{L^p(\mathbb R^n).}$$
...
- 35.5k
1
vote
1
answer
135
views
Integrability of $\exp\left(p\int_0^t |w(s,x(s,y))| \mathrm{d}s\right)$ for $w\in L^\infty(0,T;BMO(\mathbb{T}^d))$
Let $w\colon [0,T]\times\mathbb{T}^d \to \mathbb{R}^n$ be such that
$$ \|w\|_{L^\infty(BMO)} := \sup_{t\in[0,T]}\|w(t,\cdot)\|_{BMO} \leq C $$
and $\int_{\mathbb{T}^d} w(t,x)\mathrm{d}x = 0 $ for all $...
- 981
0
votes
0
answers
152
views
Predual of $BMO(\mathbb{T}^d) $
In 1971, Fefferman characterized the predual of $BMO(\mathbb{R}^d)$ as the Hardy space $H^1(\mathbb{R}^d)$.
Is there a characterization of the predual of $BMO(\mathbb{T}^d$)?
- 11
0
votes
1
answer
177
views
Irreducible sub-modules of $\ell^2(\mathbb{Z})$
It is known that $\ell^2(\mathbb{Z})$ is $\ell^1(\mathbb{Z})$-module (the module operation is the convolution).
What about the irreducible submodules? Can we characterize them?
- 1,090
2
votes
0
answers
350
views
What is the explicit version of the Peter Weyl Theorem?
While the name "Peter-Weyl" is reserved for the compact group case, I prefer to talk in greater generality. Let $G$ be a unimodular type I topological group with a fixed Haar measure. The ...
- 25.8k
1
vote
0
answers
62
views
Properties of the Fourier Transform of Countably Supported Functions on $[0,1)$
Identifying $\mathbb{R}/\mathbb{Z}$ with the interval $\left[0,1\right)$, let $C_{\textrm{coun}}\left(\mathbb{R}/\mathbb{Z}\right)$ denote the set of all functions $f:\mathbb{R}/\mathbb{Z}\rightarrow\...
- 1,284
2
votes
1
answer
382
views
Is the translation/dilation of an $L^p$-multiplier again an $L^{p}$-multiplier?
Suppose that $m:\mathbb R \to \mathbb C$ satisfies: there exists $C > 0$ such that
$$\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}.$$
That is, $m$ is an $L^{p}$-multiplier. Let $M(L^{p}...
- 63.9k
4
votes
0
answers
204
views
Harmonic functions in upper half plane
Let $\mathbb H^+$ denote the upper half plane in $\mathbb R^2$. Consider the following equation
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
\Delta u=0\,\quad &\text{on $\mathbb H^+$},...
- 4,135
4
votes
1
answer
362
views
Approximating compactly supported $L^2$ functions with Schwartz functions "from within"?
It is well known that the class of Schwartz functions $\mathcal{S}$ in dense in all $L^p$ spaces therefore for each $f \in L^2$ there exists a sequence of Schwartz functions $(f_k)$ such that $\lVert ...
- 1,869
1
vote
1
answer
228
views
Discrete harmonic analysis with infinite/unbounded number of variables
Is there any study of harmonic analysis for Boolean functions of the form $f:\{0,1\}^*\to \{0,1\}$, or $f:\{0,1\}^\omega\to \{0,1\}$?
That is, similar notions to standard harmonic analysis of $\{0,1\}^...
- 3,946
1
vote
0
answers
74
views
Dimension dependence: boundedness result of the fractional Riesz integral
I am looking for the best known constant in the boundedness result of the fractional Riesz integral. In particular, I am interested in the dependence on the dimension $d$ and on the parameter $\alpha&...
- 63.9k
1
vote
1
answer
503
views
A harmonic function $\varphi$ with $D\varphi \in L^q(\mathbb R^n)$ is constant
Let $\varphi$ be an harmonic function such that $D\varphi \in L^q(\mathbb R^n)$ for $q \in (1, +\infty)$. I read in Partial Differential Equations of Quin Han and Fanghua Lin that for $q = 2$, $\...
- 1,263
1
vote
0
answers
122
views
Metric transforms that preserve $\ell^1$ embeddability
Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances.
Work by Schoenberg and ...
- 63.9k
1
vote
0
answers
45
views
Generalizations of the Wiener Tauberian Theorem to Musielak-Orlicz spaces
Musielak-Orlicz spaces provide a generalization of the usual $L^p$ spaces on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to spaces of functions for which the Luxemburg norm
$$
\|f\|_M:=\inf\left\{\lambda &...
- 5,405
3
votes
0
answers
95
views
Sparse perturbation
Let $x, x_0\in\mathbb{R}^n$ be two vectors satisfying $$\frac{\|x\|_1}{\|x\|_2}\leq\frac{\|x_0\|_1}{\|x_0\|_2}.$$
$\| \cdot\|_1$ and $\| \cdot\|_2$ are the $\ell_1$ and $\ell_2$ norm in $\mathbb{R}^n$,...
- 31
4
votes
0
answers
146
views
Poisson summation formula for infinite dimensional spaces
Let $M$ be an orientable, compact smooth manifold with a metric $g$ and $H^{-1}(M)$ be the dual space of $$H^{1}(M)=\{f:\int |f|^2+(\nabla f)^2 d\mu<\infty\}$$
I know it is well known that (see ...
- 6,249
3
votes
1
answer
303
views
The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$
The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$ is defined by:
$$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$
where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...
CommunityBot
- 1
3
votes
0
answers
217
views
Hardy Littlewood maximal function bounds
Let $u \in W^{1,p}(\mathbb{R}^n) \cap L^{\infty}(\mathbb{R}^n)$ be a given function for some $1<p< \infty$ and let $k \in \mathbb{R}$ be any number and consider the following maximal function
$$
...
- 455
1
vote
0
answers
67
views
Approximate identities on the unit disk and going beyond a power series' radius of convergence
Let $\left\{ a_{n}\right\} _{n\geq0}$ be a bounded sequence of complex numbers, so that the power series $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ has a radius of convergence $\geq1$. ...
- 63.9k
4
votes
1
answer
221
views
Is a specific product function orthogonal to all harmonic functions
Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
- 61.9k
7
votes
0
answers
3k
views
Definition of homogeneous Sobolev spaces
As we know the inhomogeneous Sobolev space (we only consider $s>0$)
$${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ...
- 633
2
votes
0
answers
120
views
Hilbert transform on a Besov space
Consider the usual Hilbert transform of periodic functions
$$H(f) = \frac{1}{2\pi}P.V.\int_{-\pi}^{\pi}\cot(\frac{x-y}{2})f(y)dy.$$
We know $H$ does not map $L^\infty$ continuously to $L^\infty$. Now ...
- 903
7
votes
1
answer
1k
views
Where does the Laplace transform come from?
The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
- 107
2
votes
0
answers
158
views
Estimate involving Besov norm
When reading some old notes of my advisor on interpolation spaces, I bumped into a problem I can't quite wrap my head around. Here are the details.
For $p\in(0,\infty)$ a $p$-variation semi-norm of a ...
- 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_\...
- 17.2k
13
votes
1
answer
675
views
Wavelet-like Schauder basis for standard spaces of test functions?
Edit: A more precise formulation of my question follows the separation line.
The Schwartz space of test functions $\mathcal{S}(\mathbb{R})$ is isomorphic to $\mathfrak{s}$ the space of sequences of ...
- 7,817
5
votes
2
answers
881
views
Fourier transform on locally compact quantum groups
I have read some articles on locally compact quantum groups and the Fourier Transform on them. I wonder why we define the Fourier transform as an operator valued functions from $L^1(\mathbb{G})$ to $L^...
- 63.9k
3
votes
0
answers
342
views
A.C.M. van Rooij's *Non-archimedean functional analysis* (1978) is very out-of-print! Anyone know of any good alternatives?
(This is a literature/reference question.)
So... long story short:
(1) In my present research, I needed a theory of continuous functions from the $p$-adic integers to the $q$-adic integers. Unable ...
- 1,284
2
votes
0
answers
221
views
Besov or Triebel-Lizorkin spaces versus Lorentz spaces
I first asked this question on math.stackexchange here but it seems it is more a research level question ...
At the $0$ order of derivatives of Sobolev spaces and for a fixed integrability order $p$, ...
- 230
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 $f(0-)$...
- 91.8k