All Questions
Tagged with fa.functional-analysis harmonic-analysis
190 questions with no upvoted or accepted answers
3
votes
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206
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Explicit basis of symmetric harmonic polynomials
An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki.
From there, constructing an orthonormal basis for ...
- 31
3
votes
0
answers
84
views
Convergence of the Gaussian integral on $\mathcal{E}'$ for a mapping supported on $L^2$
Let $F : L^2(S^1) \to L^2(S^1)$ be a (nonlinear) mapping such that
\begin{equation}
\lVert F(f) \rVert \leq \lVert f \rVert
\end{equation}
for all $f \in L^2(S^1)$. For the space of smooth periodic ...
- 3,477
3
votes
0
answers
162
views
The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 4,058
3
votes
0
answers
77
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Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$
My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
- 41
3
votes
0
answers
217
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Hardy Littlewood maximal function bounds
Let $u \in W^{1,p}(\mathbb{R}^n) \cap L^{\infty}(\mathbb{R}^n)$ be a given function for some $1<p< \infty$ and let $k \in \mathbb{R}$ be any number and consider the following maximal function
$$
...
- 455
3
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0
answers
342
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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
3
votes
0
answers
192
views
Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$
Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform.
...
- 230
3
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0
answers
164
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On Pitt's inequality (weighted Fourier inequality)
One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$,
$$
\sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(...
- 1,216
3
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answers
151
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Completeness of discrete shifts in $\mathbb{R}^n$
Consider the space $L^2(\mathbb{R})$. Let $(x_n)_n \subset \mathbb{R}$ be a sequence and $f \in L^2(\mathbb{R})$ a functions. It is well understood under which assumptions the span of the set
$$
S = \{...
- 173
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
3
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0
answers
180
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When is a minimal immersion holomorphic?
Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let:
$\phi\colon X\to Y$
be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am ...
- 2,816
3
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answers
168
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Zak transform and VMO
The Zak transform of a function $f\in L^1(\mathbb R)\cap L^2(\mathbb R)$ is defined as follows:
$$
Zf(x,\omega) := \sum_{k\in\mathbb Z}f(x+k)e^{-2\pi i k\omega},\quad (x,\omega)\in Q_0 :=(0,1)^2.
$$
...
3
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0
answers
126
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An identity of operator norms and de Leeuw's theorem
Let $$Hf(x_1,x_2)=p.v.\int_{-\infty}^\infty f(x_1-t,x_2-S(x_1,x_1-t))\frac{dt}{t},$$
$$T_\lambda f(x)=\lim_{\epsilon\to0}\int_{|x-y|\ge\epsilon}e^{i\lambda S(x,y)}(x-y)^{-1}f(y)dy, $$ where $S(x,y)$ ...
- 187
3
votes
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214
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Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698
3
votes
0
answers
317
views
Best constant for maximal function for locally compact groups
Maximal functions for locally compact groups have been studied. In particular papers of K. Phillips and Phillips-Taibleson (here and here) contain nice results in this direction. It might be too much ...
- 1,583
3
votes
0
answers
262
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Non-compact analogue of Peter-Weyl
I have the following situation: $G$ is a real unimodular locally compact semisimple Lie group. Then it is known that the regular representation $H:=L^2(G,\mu_H)$ decomposes as
\begin{equation}
\int^{\...
- 1,813
3
votes
0
answers
211
views
Arveson spectrum for a unitary representation of a group on a Hilbert space
Although this is not research, I think the question is a little bit too specific for math.stackexchange
Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a ...
- 189
3
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0
answers
140
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convergence of $e^{it\Delta}f$
I heard of a conjecture that
$e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$
but couldn't find a proper reference.
- 31
3
votes
0
answers
286
views
Tauberian theorem from generalized Gelfand transform
Wiener's theorem gives the necessary and sufficient conditions for the set of translates of a set of functions to be dense in $L^1(\mathbb{R}^n)$, which translates algebraically into a statement about ...
- 1,049
3
votes
0
answers
301
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What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?
Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...
user23860
3
votes
0
answers
145
views
Growth of inner functions on the disk
Recall that an inner function on the disk $D$ is a bounded analytic function on $D$ having radial limits of modulus one almost everywhere.
There has been many works on the growth of the inner ...
- 769
2
votes
0
answers
86
views
Besov spaces containing piecewise linear functions
Let $\Omega$ be a bounded, non-empty, regular open domain in $\mathbb{R}^d$. Let $1\le p,q\le \infty$ and $s>0$. Let $\mathcal{B}_{p,q}^s(\Omega)$ be the Besov space on $\Omega$ corresponding to ...
2
votes
0
answers
30
views
Dual of homogeneous Triebel-Lizorkin
Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with
$$
[f]^{p}_{\dot{F}^{s}_{p,q}...
- 31
2
votes
0
answers
139
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Multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
How to rigorously demonstrate that multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
I am utilizing the exponential type definition established by Elias Stein in the book 'Fourier ...
- 21
2
votes
0
answers
180
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Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
2
votes
0
answers
203
views
Schrödinger representation of the Heisenberg group
Let $\Pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H^n=\Bbb C^n\times\Bbb R$. For $\phi\in L^2(\Bbb R^n)$, we have
$$\Pi_{\lambda} (x,y,t)\phi(\xi)=e^{i\lambda t} e^{...
- 501
2
votes
0
answers
206
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Failure of Calderón–Zygmund inequality at the endpoints
$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'd like to prove that the famous Calderón–Zygmund elliptic estimate $$ \norm{ \partial_{ij}u }_{L^p} \leq C \norm{\Delta u }_{L^...
- 457
2
votes
0
answers
88
views
Explicit estimates on summability kernels
A "summability kernel" is a sequence of functions $k_n:[0,1)\to \mathbb C$ such that
$$ \int_0^1 k_n(t) \mathrm d t =1,$$
$$ \int_0^1 |k_n(t)| \mathrm d t =O(1),$$ with an implied constant ...
- 3,062
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 ...
- 455
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 $\mathcal S(\mathbb R^8)$
$$ V(...
- 73
2
votes
0
answers
81
views
An square root of the multiplicative operator on $\ell^1(\mathbb{Z}_n)$
Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define
$$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$
...
- 4,058
2
votes
0
answers
149
views
A closed ideal in $L^1(T)$
Let $\mathbb{T}$ be the unit circle and consider the convolution group algebra $L^1(\mathbb{T})$. Let $I_n$ be the closed ideal generated by the polynomial $p_n(z)=z^n-1$ in $L^1(\mathbb{T})$.
Let $I=...
- 4,058
2
votes
1
answer
547
views
Shift-invariant spaces
We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...
- 49
2
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0
answers
117
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Is the cone of positive elements in $L^1(G)$ norm closed?
Let's consider $L^1(G)$, the Banach $*$- algebra of all Lebesgue integrable functions on the locally compact group $G$. Put $L^1(G)_+$ by the cone of positive elements given by $\{\sum_1^n f_i^**f_i: ...
- 4,058
2
votes
0
answers
190
views
Inequality on the dual space of $H^s$
Does there exist a theorem that allow us to say that, if we have an estimate on the Sobolev space $H^s\,,\, s\geq 0$ then we can deduce an estimate on the dual space $H^{-s}$ ?
For instance, assume ...
- 93
2
votes
0
answers
120
views
Does a holomorphic function with logarithmic growth at the boundary have $L^2$ boundary values?
Let $f(z)$ be a holomorphic function on the unit disc, with logarithmic growth at the boundary:
$$
|f(z)| = \mathcal O\bigg(\log\Big(\frac{1}{1-|z|}\Big)\bigg).
$$
Does it follow that the (...
- 43.2k
2
votes
0
answers
164
views
(Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis
It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
- 1,245
2
votes
0
answers
191
views
Convergence in $S'(\mathbb R^d)$ of the paraproduct $\dot{T}_uv$
Let $B = B(0,4/3)$, $C = \{x \in \mathbb R^d : 3/4 \leq \|x\|_2 \leq 8/3\}$ and $\tilde{C} = \{x \in \mathbb R^d : 1/12 \leq \|x\|_2 \leq 10/3\}$.
For a fixed Littlewood-Paley decomposition $\chi \in \...
- 233
2
votes
0
answers
169
views
Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
- 839
2
votes
0
answers
349
views
What is the explicit version of the Peter Weyl Theorem?
While the name "Peter-Weyl" is reserved for the compact group case, I prefer to talk in greater generality. Let $G$ be a unimodular type I topological group with a fixed Haar measure. The ...
- 2,071
2
votes
0
answers
120
views
Hilbert transform on a Besov space
Consider the usual Hilbert transform of periodic functions
$$H(f) = \frac{1}{2\pi}P.V.\int_{-\pi}^{\pi}\cot(\frac{x-y}{2})f(y)dy.$$
We know $H$ does not map $L^\infty$ continuously to $L^\infty$. Now ...
- 903
2
votes
0
answers
158
views
Estimate involving Besov norm
When reading some old notes of my advisor on interpolation spaces, I bumped into a problem I can't quite wrap my head around. Here are the details.
For $p\in(0,\infty)$ a $p$-variation semi-norm of a ...
- 421
2
votes
0
answers
221
views
Besov or Triebel-Lizorkin spaces versus Lorentz spaces
I first asked this question on math.stackexchange here but it seems it is more a research level question ...
At the $0$ order of derivatives of Sobolev spaces and for a fixed integrability order $p$, ...
- 230
2
votes
0
answers
249
views
Links between differing notions of "pseudo-measure"'; or, why that name?
(A pet peeve of mine is Mathematicians from field X noticing that field Y uses terminology which is very close to that from field X, and assuming there are Mathematical links. This question might be ...
- 18.7k
2
votes
0
answers
171
views
How to use Stein-Tomas theorem to check to following inequality?
Recently, I am reading Rodnianski & Schlag
Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. In lemma 3.2, R&S said that by using Stein-Tomas theorem ...
- 429
2
votes
0
answers
126
views
On the infimium of a functional
Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$
where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
- 169
2
votes
0
answers
418
views
Hölder-Zygmund spaces of negative order
In the equation (1.1.17) in Proposition 1.1.6 (ii) in Alazard and Delort's Sobolev Estimates for Two Dimensional Water Waves, there appears a norm named $C^{-1}$, but in Chapter 6 (Appendix) of the ...
- 5,169
2
votes
0
answers
70
views
Can the STFT decrease arbitrarily quickly near the origin?
For $f,g \in L^2(\mathbb{R}^d)$ we can define the Short Time Fourier Transform (STFT) $V_gf \in C_0(\mathbb{R}^{2d})$ as $$V_gf(x, \omega) = \int_{\mathbb{R}^d} f \overline{g(t - x)} e^{-2 \pi i t \...
- 1,010
2
votes
0
answers
89
views
Link between subharmonic and subanalytic functions
Consider $\Omega$ an open set of $\mathbb{C}$ and $f : \Omega \to \mathbb{R}$ a $C_{\infty}$-function. The following two definitions are well-known (at least the first one) but I prefer to recall them ...
- 1,017
2
votes
0
answers
136
views
Equivalent statement of the Wiener-Tauberian theorem?
I would like to know why we have the equivalence between the following three statements of the Wiener-Tauberian theorem:
version 1: If $I$ is a closed ideal in $L^1(\mathbb R)$, such that the set $...
- 650