All Questions
389 questions
5
votes
1
answer
490
views
$L^2$ uniform integrability in terms of Fourier coefficients
Given a bounded sequence $(f_n)_n$ in $L^2(\mathbf{T})$ where $\mathbf{T}:=\mathbf{R}/\mathbf{Z}$, the strong compactness of $(f_n)_n$ is equivalent to $$\lim_N \sup_n \sum_{|k|\geq N} |c_k(f_n)|^2=0,$...
- 3,425
5
votes
0
answers
77
views
Are these two versions of Sobolev embedding related?
In Griffith-Harris Section 0.6 we have this Sobolev lemma:
Let $H_s$ be the space of formal Fourier series $u(x):=\sum_{k\in \mathbb Z^n}u_ke^{i(k,x)}$ on $(\mathbb R/2\pi\mathbb Z)^n$ such that the $...
- 1,630
5
votes
0
answers
168
views
Sobolev extension from a discrete set of points
Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define
$$...
- 501
5
votes
0
answers
262
views
Weighted reverse Poincare inequality over a function class of neural networks
We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
- 325
5
votes
0
answers
120
views
Geometric characterization of Silva distributions
There is a well known geometric characterization of tempered distributions on $\mathbb{R}^n$.
A distribution $T\in \mathcal{D}'(\mathbb{R}^n)$ is an element of $\mathcal{S}'(\mathbb{R}^n)$ if and ...
- 1,017
5
votes
0
answers
166
views
Fourier basis for sub-Gaussian spaces?
Let $(\mathcal{X}, \pi)$ be a probability space such that $\pi$ has full support. Consider $L^2(\mathcal{X},\pi)$ to be the inner product space of function $f: \mathcal{X}^n \to \mathbb{R}$, with ...
- 519
5
votes
0
answers
913
views
Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
- 103
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
4
votes
2
answers
1k
views
Can we extract information about how fast a function decay from its Laplace transform?
My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.
More concrete case, let $f:\mathbb{R} ...
- 1,839
4
votes
5
answers
3k
views
Generalize Fourier transform to other basis than trigonometric function
The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.
In how far can this ...
- 5,327
4
votes
1
answer
289
views
Is there a name for this space?
I'm just asking if there is a name for the space of functions on $\mathbb R^n$ whose norm is defined by
$$ \|f\|=\|\hat f\|_{L^p} $$
for $p\in [1,\infty]$. I find it handy to give it a name when ...
- 5,169
4
votes
1
answer
497
views
About the boundedness of a multiplication operator.
Let be $f$ a $2\pi-$periodic function and $\hat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}dx$. Consider the operator:
\begin{equation}
Tf(x)=\sum_{k\in\mathbb{Z}}sign(k)\ \hat{f}(k)\ e^{ikx}.
\end{...
- 45
4
votes
1
answer
214
views
Equivalent Littlewood-Paley-type decompositions
The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that
\begin{...
- 989
4
votes
2
answers
1k
views
Characterizations of Wiener algebra
The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that
$$
\mathcal W\subset ...
- 16.2k
4
votes
1
answer
596
views
Modulus of of continuity of a convolution operator with respect to Wasserstein metric
For a (discrete) measure $G$ on some reasonable metric space $\Theta$, consider the map $G \mapsto f_G$ defined as
$$
f_G := f*G(dx) := \int f(dx|\theta) G(d\theta)
$$
for some nice kernel function $...
- 1,731
4
votes
1
answer
277
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
4
votes
1
answer
203
views
If $f$ is non-prime, can we say $|f|$ is also a non-prime; in convolution algebra?
By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that,
$$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): \sum_{n\in \mathbb Z} |\hat{f}(n)| < \...
- 1,051
4
votes
1
answer
471
views
Ask for theory about the weighted L^2(R^d) space.
Dear MOs,
I am now considering the following norm:
$$
||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:.
$$
where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
- 1,649
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(\...
4
votes
1
answer
2k
views
Characterizations of a linear subspace associated with Fourier series
Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace
a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable ...
- 744
4
votes
1
answer
255
views
Proof that elements of Beppo-Levi-like spaces are functions (and not just distributions)?
Context. I am trying to undestand the theory underlying "Beppo-Levi"-like spaces defined as
$$
H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{...
4
votes
1
answer
253
views
Regarding outer functions
Please see the definition of Hardy spaces on the unit disc here. Let $0<p\leq\infty$. Let $f\in H^p$ with $\|f-1_e\|_p<1$ (Where $1_e$ Is the constant function one). Then is $f$ an outer ...
- 111
4
votes
1
answer
277
views
Does the Fourier transform preserve the separation property?
The space of Schwartz functions on the plane is denoted by $\mathcal{S}$.
The usual multiplication and the convolution multiplication on $\mathcal{S}$ are denoted by $m_1$ and $m_2$, respectively.
...
- 356
4
votes
0
answers
158
views
Measurability of $L^{p}(L^{q})$ integrable functions
Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that
$
\int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty
$
In addition we ...
- 41
4
votes
0
answers
140
views
Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
- 1,075
4
votes
0
answers
310
views
Fourier characterization of weighted Sobolev space $W^{1,2}(\mathbb R^n, \gamma_n)$
For integers $n \ge 1$ and $m \ge 0$, the Sobolev space $W^{m,2}(\mathbb R^n)$ is characterized by
$$
f \in W^{m,2}(\mathbb R^n) \text{ iff } \tilde f_m \in L^2(\mathbb R^n),
\label{1}\tag{1}
$$
where ...
- 6,853
4
votes
0
answers
81
views
Does this sequence of functions converge in a distributional sense?
Let $f\in W^{1,12/5}(\mathbb{R}^3)$ (time-independent), let $K^{\epsilon}$ be a uniformly in $\epsilon$ bounded sequence in $L^{1}\cap L^{7/5}(\mathbb{R}^3)$ and let
$$\tilde{K}^{\epsilon} := K^{\...
- 469
4
votes
0
answers
255
views
Pointwise convergence of kernels of Hilbert-Schmidt operators
Lately I was discussing different types of convergence for Hilbert-Schmidt operators and during that discussion we ended up talking about pointwise convergence of Fourier series. I have already asked ...
- 1,045
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
4
votes
0
answers
205
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 $\mathbb H^+$},...
- 4,143
4
votes
0
answers
444
views
Smoothness and decay correspondence for Laplace transform
For the Fourier transform, there are various theorems formalizing a correspondence between the smoothness of a function and the rate of decay of its Fourier transform. For example, if a function is $n$...
- 5,827
4
votes
0
answers
207
views
Simultaneous Hahn-Banach theorem
Let $C(\mathbb{T})$ be the Banach algebra of continuous functions on the unit circle. Let $n \in \mathbb{N}$ and let $P_n(\mathbb{T})$ be the subspace of trigonometric polynomials of degree at most $n$...
- 1,769
4
votes
0
answers
965
views
Norms of the Dirichlet kernel
I guess that the following estimates are classical. Let $D_N$ be the $1D$ Dirichlet kernel,
$$
D_N(t)=\frac{\sin((N+\frac12)t)}{\sin (t/2)}.
$$
We have for $1<p<\infty$,
\begin{align}
\Vert D_N\...
- 16.2k
4
votes
0
answers
116
views
Is there a categorical foundation for manifolds of bounded geometry and bandlimited functions?
As an outsider to both, manifolds of bounded geometry and bandlimited functions appear rather connected: for example, bounded geometry is defined in terms of bounds on curvature and its derivatives, ...
- 3,612
4
votes
0
answers
2k
views
Fourier transform of $C^\infty_0$, smooth functions vanishing at infinity
Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
- 41
4
votes
0
answers
551
views
$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?
Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra
$$A(\mathbb R):= \{f\in L^{1}(...
- 1,051
4
votes
1
answer
167
views
A Laplacian semi-group estimation
Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by
$$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
- 43
3
votes
2
answers
279
views
Nice way to express $H^{-1}(\mathbb{S}^1)$
I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
- 33
3
votes
2
answers
589
views
On the Fourier inversion formula
For a given function $f\in L^1(\mathbb{R})$, suppose that the
$$\check{f}(x)=\int_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$
almost every where converges in $\mathbb{R}$. Then, can we say that
...
- 4,058
3
votes
1
answer
480
views
Is there a uniform upper bound for this oscillatory integral?
I am wondering whether the following uniform upper bound holds:
$|\int_a^{2a}\frac1t\sin(N b^2t)\exp(iNbt^2)dt|\le Cab^2,$
where $0<a<b<1$, $N>N_0(a,b)\gg1$, and $C$ is a constant ...
- 187
3
votes
2
answers
1k
views
A sufficient condition for a probability measure to have compact support
Consider a probability measure $\mu$ on, let's say, $\mathbb R$.
Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ?
I agree this question is too vague, ...
- 2,135
3
votes
1
answer
182
views
How to choose some $h$ so its Fourier transform supported in some set?
Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
- 325
3
votes
2
answers
869
views
How do functions operate in a Sobolev space $H^{s}$?
Let $s>\frac{1}{2};$ and define a Sobolev space as follows:
$$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$
Fact: Let $m$ ...
- 1,051
3
votes
1
answer
195
views
Boundedness of different Fourier transforms
Let $f: \mathbb{R}^n \rightarrow \mathbb{C}$ be in $L^2\cap L^1,$ then the Fourier transform is in $L^2 \cap L^\infty.$
Does this imply that we can take common norms in the sense that we can estimate ...
- 33
3
votes
1
answer
296
views
Uniform convergence of generalised Fourier series
Suppose $u_n$ is an orthonormal basis of smooth functions on $S^1$.
Does there exist a smooth function $u$ such that the generalised Fourier series
$$u=\sum_{n\in\mathbb{N}} \langle u,u_n\rangle u_n ...
- 155
3
votes
1
answer
336
views
What are the almost periodic functions on the complex plane?
The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally ...
- 173
3
votes
2
answers
1k
views
Fourier transform inversion theorem for a function not in L1 or L2
For $\frac{1}{4}<a<1$ consider the following function:
$$f(x)=\frac{|x|^{\frac{1}{2}}}{(x^2+1)^{a+ib}}$$
If $1>a>\frac{1}{2}$ then $f(x) \in L^2$ and the Fourier inversion theorem can be ...
- 1,199
3
votes
3
answers
580
views
Approximate identities and pointwise convergence
I'm studying Fourier analysis and have a question about approximate identities.
Let $k_{\epsilon}$ be an approximate identity on $L^{1}(\mathbf{T})$. We know that $k_{\epsilon}*f\to f$ in $L^{1}$ as $...
- 41
3
votes
1
answer
416
views
Let $f \in S(\mathbb R)$. Can we say $\widehat{|f|} \in L^{1}(\mathbb R)$?
Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int_{\mathbb R} | f(x)| dx < \infty \}$ and the Fourier transform of $f$,
$\hat{f} (y) : = \...
- 1,051
3
votes
2
answers
477
views
Vanishing convolution between density and compactly supported function
Find a pair of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that:
$f$ is smooth and compactly supported (say, on $[0,1]$ but this isn't crucial),
$g(x)>0$ for all $x\in\mathbb{R}$, $\int g(x)\,dx=...
- 75