All Questions
Tagged with fa.functional-analysis harmonic-analysis
451 questions
2
votes
0
answers
99
views
Anisotropic Calderon-Zygmund decomposition
I am looking for the following version of Calderon-Zygmund decomposition, consider an function $f \in L^1(R^{d+1})$ and cylinders of the form $Q_{R,R^p}$ for some fixed $p \in (0,\infty)$, The ...
- 63.9k
1
vote
0
answers
169
views
Almost everywhere convergent Fourier series
Apparently there is a deep theorem stating:
Let $f:\mathbb{R}\to \mathbb{C}$ be a function satisfying $f(x)=f(x+2\pi)$ and $\int_0^{2\pi}|f(x)|^2dx<\infty$. Then the Fourier series of $f$ converges ...
- 63.9k
4
votes
0
answers
68
views
Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$
For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that
$$
|f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
- 61
0
votes
0
answers
75
views
$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
- 159
1
vote
0
answers
47
views
Holomorphic "quasi-interpolation" of a function sequence
I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...
- 981
8
votes
2
answers
1k
views
What does the unique mean on weakly almost periodic functions look like?
There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...
- 426
2
votes
1
answer
261
views
Qualitative difference between "continuous" and "discontinuous" states on $M(G)$
Let $G$ be a locally compact Abelian group (we can think that $G={\mathbb R}$). Let $C_0(G)$ be the space of continuous functions $u:G\to{\mathbb C}$ vanishing at infinity with the usual $\sup$-norm, ...
- 18.7k
12
votes
1
answer
1k
views
Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$
Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...
- 2,821
0
votes
0
answers
72
views
Fourier coefficient of close functions
Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as
$$ f(x) = \...
- 63.9k
1
vote
0
answers
142
views
Splitting of a topological vector space (TVS) into an (a) countable sum and (b) direct integral of subspaces
I thought that this would be a simpe question, and placed it here at the Mathematics Stackexchange. Now have to elevate it to Mathoverflow.
LANGUAGE
TVS = topological vector space. Any subspace of a ...
- 603
1
vote
0
answers
147
views
BMO estimates of singular integral operators on torus
I have the following elliptic problem:
$$ \Delta u = \operatorname{div}\operatorname{div}S, $$
where $S=(S_{i,j})\colon \mathbb{T}^n\to \mathbb{R}^{n\times n} $ is bounded and $\mathbb{T}^n$ is the $n$...
- 63.9k
2
votes
0
answers
216
views
Fourier transform of Dirac delta distribution
Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$
$$ V(...
- 73
1
vote
0
answers
64
views
Characterization of elements of Hardy Space
Let $\Omega\subset\mathbb{C}^n$ be a $C^{\infty}$ bounded domain. Let $H^2(\partial\Omega)$ denote the Hardy space, and $S(.,.)$ denote its Szego Kernel. We know that
$$
\forall f\in H^2(\partial\...
- 4,713
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)(=\...
- 25.8k
2
votes
0
answers
81
views
An square root of the multiplicative operator on $\ell^1(\mathbb{Z}_n)$
Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define
$$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$
...
- 4,058
3
votes
1
answer
153
views
Urysohn's lemma for Bochner functions?
Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used:
If $U$ is an open ...
- 175
5
votes
1
answer
281
views
de Rham theorem for tempered distributions
I am wondering if the following statement holds.
If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...
- 24.2k
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 ...
- 23k
4
votes
0
answers
161
views
Hodge theory in higher eigen-spaces?
Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology
$$\mathcal{H}(E) \simeq H(E).$$
A classical example with differential forms ($E = (\Omega,d)$) ...
- 5,230
3
votes
1
answer
194
views
Analytic approximation of the step function in $L^p$ norm
Motivation: Euler-Maclaurin formula uses calculus to estimate discrete sums. I wonder what one can do by reverse engineering. A concrete problem I ran into is the following.
Question: Let $\chi: \...
- 41.1k
2
votes
0
answers
149
views
A closed ideal in $L^1(T)$
Let $\mathbb{T}$ be the unit circle and consider the convolution group algebra $L^1(\mathbb{T})$. Let $I_n$ be the closed ideal generated by the polynomial $p_n(z)=z^n-1$ in $L^1(\mathbb{T})$.
Let $I=...
- 4,058
3
votes
0
answers
77
views
Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$
My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
- 63.9k
3
votes
1
answer
667
views
Equivalent action of convolution of directional derivative
I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented.
Let $k*f(x)...
- 350
1
vote
0
answers
57
views
How to show the solution map of NLS is not smooth?
Let $u(\delta, t)$ satisfy
$$iu_t +\Delta u+ |u|^{2k}u=0, \quad u(0)=\delta v_0$$
Note that the mapping:
$$\delta v_0\mapsto u(\delta, t)= S(t)(\delta v_0)-i\int_0^tS(t-\tau)(|u|^{2k}u)(\tau)d\tau $$
...
- 325
10
votes
2
answers
594
views
Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?
Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
- 18.7k
9
votes
2
answers
1k
views
Density of restrictions of harmonic functions inside a ball
Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let
$$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
1
vote
0
answers
79
views
A problem arising from Wiener-Levy theorem on the real line
Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...
- 4,058
1
vote
0
answers
53
views
Stability of Hajłasz-Sobolev class under post-composition
Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev?
Assumptions/Setup
Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
- 5,405
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 ...
- 1,848
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 ...
- 128k
2
votes
0
answers
117
views
Is the cone of positive elements in $L^1(G)$ norm closed?
Let's consider $L^1(G)$, the Banach $*$- algebra of all Lebesgue integrable functions on the locally compact group $G$. Put $L^1(G)_+$ by the cone of positive elements given by $\{\sum_1^n f_i^**f_i: ...
- 4,058
4
votes
1
answer
221
views
Fourier multipliers and transference on cyclic groups
It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on $L^p(\mathbb{R}^n)$ then by transference the similar operator is also bounded on $L^p(\...
5
votes
1
answer
496
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
0
answers
190
views
Inequality on the dual space of $H^s$
Does there exist a theorem that allow us to say that, if we have an estimate on the Sobolev space $H^s\,,\, s\geq 0$ then we can deduce an estimate on the dual space $H^{-s}$ ?
For instance, assume ...
- 6,367
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
1
vote
0
answers
260
views
Closure of smooth functions in Besov spaces
For real numbers $\alpha > \beta$, we know there is a continuous embedding of Besov spaces $B^\alpha_{\infty,\infty}\subset B^\beta_{\infty,\infty}$. We take the closure of the intersection $C^{\...
- 253
0
votes
0
answers
84
views
Determining the tails of a convolution from its behavior on a compact set
Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
- 19
2
votes
0
answers
120
views
Does a holomorphic function with logarithmic growth at the boundary have $L^2$ boundary values?
Let $f(z)$ be a holomorphic function on the unit disc, with logarithmic growth at the boundary:
$$
|f(z)| = \mathcal O\bigg(\log\Big(\frac{1}{1-|z|}\Big)\bigg).
$$
Does it follow that the (...
- 6,367
11
votes
1
answer
413
views
Estimating the growth of the Taylor coefficients given the growth of the function at the boundary
Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies
$$
|f(z)|\le \frac{1}{(1-|z|)^{k}}
$$
for some fixed $k>0$.
Question: What can I deduce about the growth of the ...
- 43.2k
3
votes
1
answer
171
views
Discrete singular integrals
Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:
$\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
- 128k
-1
votes
1
answer
78
views
Fundamental of a signal
Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$
or any reasonable Euclidean norm such that bounded ...
- 26.3k
1
vote
1
answer
518
views
On level sets of smooth functions in a bounded domain
Let $\Omega$ be a bounded domain in $\mathbb R^n$, $n\geq 2$, with a smooth boundary and let $f$ be a smooth function on $\bar\Omega$. Is there a natural condition that one can impose on $f$ ( say in ...
- 91.8k
1
vote
0
answers
244
views
On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)
Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
- 111
4
votes
0
answers
104
views
Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$
Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
- 6,367
4
votes
0
answers
127
views
Algebra properties regarding Gevrey spaces: closed under multiplication
In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
- 517
3
votes
1
answer
164
views
Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?
Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may ...
- 2,504
13
votes
0
answers
395
views
Converse to Riesz-Thorin Theorem
Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.
The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....
- 854
0
votes
0
answers
85
views
An amenable operator algebra has the total reduction property
This is from
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CB20539885C03522D141C34024707702/S1446788700014026a.pdf/div-class-title-operator-algebras-with-a-reduction-property-...
- 101
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
12
votes
1
answer
727
views
A generalization of Rubio de Francia's inequality
Suppose that $\{I_m\}$ is a sequence of pairwise disjoint intervals in $\mathbb{Z}$. The well known Rubio de Francia's inequality says that for any function $f\in L^p(\mathbb{T})$, $2\le p<\infty$, ...
- 114k