All Questions
389 questions
2
votes
0
answers
109
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
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
2
votes
0
answers
79
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
1
vote
1
answer
294
views
Regarding subspace generated by the polynomial multiples of outer functions
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
CommunityBot
- 1
1
vote
0
answers
86
views
Density of a subset of Schwartz space in the fractional Sobolev space
It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$, (where $0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}...
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,713
2
votes
0
answers
194
views
Functions such that the *integral* of the Fourier transform is non-negative?
Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that
$$\int_{-\infty}^x \widehat{f}(t) dt \...
- 20.2k
3
votes
3
answers
581
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 $...
1
vote
0
answers
87
views
Proof mistake of: $M_0A(G) = B(G)$ for a locally compact group
I am posting my question of mathstack exchange here. (see: My post on MSE)
Let $G$ be a locally compact group with Haar measure $\mu$, and $B(G),A(G),C_r^*(G),L(G)$ be its Fourier-Stieltjes algebra, ...
- 261
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(\...
CommunityBot
- 1
5
votes
1
answer
320
views
Is $\mathscr{S}_h'$ a complementary subspace for $\mathscr{S}'/\mathscr{P}$, the space of tempered distributions modulo polynomials?
Recall that in many Fourier Analysis texts, given a function $\Psi$ such that $\hat{\Psi}\in\mathcal{D}(\mathbb R^d)$, $\hat\Psi\ge0$ is supported in an annulus, and $\sum_{j\in\mathbb Z}\hat\Psi(2^j\...
- 1,247
1
vote
1
answer
128
views
Infinite direct sum decomposition of the heat semigroup on $\mathbb{R}^n$
Consider the heat semigroup $Q_t$ on $L^2(\mathbb{R}^n)$ generated by the Laplace operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x^2_i}$. Does there exist a direct sum decomposition
$$\oplus_{...
- 6,476
0
votes
1
answer
117
views
Validity of approximation method for von Mangoldt function
I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at ...
CommunityBot
- 1
1
vote
1
answer
127
views
approximating differentiable functions with double trigonometric polynomials
Let $Q = [0,1]^2$. For sake of notation, let
$$
f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi).
$$
Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if
$$
\|...
CommunityBot
- 1
1
vote
1
answer
130
views
Existence of solutions to a series of integral equations
I am trying to solve the following integral equation analytically:
$$
\sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T],
$$
where $(f_n(t))_n$ is the unknown ...
CommunityBot
- 1
7
votes
2
answers
824
views
Fourier series of smooth functions in infinitely many variables
Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
CommunityBot
- 1
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 $$\lim_N \sup_n \sum_{|k|\geq N} |c_k(f_n)|^2=0,$...
CommunityBot
- 1
23
votes
0
answers
1k
views
Laplace Transform in the context of Gelfand/Pontryagin
Questions:
Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform?
If not, is there a ...
- 1,124
11
votes
2
answers
8k
views
About the Fourier transform of the logarithm function
I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
- 201
0
votes
1
answer
127
views
Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w) < \infty$?
Let $f\in L^p(\Bbb R^n)$ and $S^{n-1}$ be the Unit sphere.
Why
$\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w)<\infty$ when $1<p<2$.
$\hat{f}$ is the Fourier transform fora function f.
- 39k
1
vote
0
answers
105
views
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
- 12.9k
9
votes
2
answers
758
views
Number of critical points of smooth functions on $S^1$
Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?
- 18.7k
1
vote
1
answer
111
views
How to show such result for generalized $ O(|x|^{-1/2}) $ function?
Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
- 24.2k
0
votes
0
answers
89
views
Maximal function on mixed $L^{p}$
Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is
$$
\Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
- 31
0
votes
1
answer
80
views
Orthogonal space of polynomials
Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and
$$
\int_0^{+\infty} f(t) t^n ...
- 3,425
0
votes
1
answer
126
views
Clarification on the Interpretation of Fourier Coefficients in the Context of Fourier Projections
I am currently studying a paper (Section 3.4.3 of Lanthaler, Mishra, and Karniadakis - Error estimates for DeepONets: a deep learning framework in infinite dimensions) where the authors define an ...
- 7,817
3
votes
0
answers
100
views
Rate of convergence of mollified distributions in Besov spaces with negative regularity
Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
- 293
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
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 ...
- 31
6
votes
2
answers
458
views
Does the (distributional) support of the Fourier transform of an $L^p$-function with $p<\infty$ have positive measure?
Suppose that $f \in L^p(\mathbb R^n)$ such that $1\leq p < \infty$. Let $\hat f$ be the Fourier transform of $f$. Clearly, if $p=1$ or $p=2$ then the support of $\hat f$ has positive Lebesgue ...
0
votes
0
answers
53
views
Vectors of complex exponentials span $\mathbf{C}^N$
Let $Q = [0,1]\times [0,1]$. Let $1\leq k < \infty$ and $\{(x_l,\xi_l)\}_{l=1}^{k}\subseteq Q$ be such that $(x_i,\xi_i)\ne (x_j, \xi_j)$ for $i\ne j$. Additionally, for $1\leq l \leq k$, let $n_l\...
- 63.9k
1
vote
1
answer
121
views
An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 6,476
0
votes
1
answer
102
views
On weighted Fourier transforms
Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that
$$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$
...
- 6,476
1
vote
1
answer
112
views
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
5
votes
1
answer
246
views
An asymmetric quadrilinear estimate
Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$
where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
- 61.9k
0
votes
1
answer
121
views
A simple bilinear estimate
Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that
$\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$.
Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$.
What is the optimal value of $t=t(\...
- 852
2
votes
1
answer
320
views
Fourier series but different waveform
Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
3
votes
1
answer
162
views
Rescaling Fourier coefficients of a continuous function by a bounded sequence
This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
- 24.2k
44
votes
10
answers
47k
views
Is square of Delta function defined somewhere?
I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, ...
2
votes
1
answer
223
views
Sobolev regularity via Laplace spectrum
Fix a positive integer $n$ and let $\mu$ be the uniform measure on the sphere $\mathbb{S}^n$, with respect to its usual Riemannian metric $g$. Let $\nabla$ be the Laplacian on $(\mathbb{S}^n,g)$ and ...
- 23k
0
votes
0
answers
94
views
The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
- 63.9k
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{...
- 484
9
votes
1
answer
639
views
Prove J.L. Lions’s Lemma without using Fourier transform
When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...
- 188k
2
votes
0
answers
139
views
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
8
votes
1
answer
496
views
A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
- 6,035
0
votes
1
answer
507
views
Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 684
9
votes
1
answer
916
views
Inverse Fourier transform of an $L^2$ function as limit on balls
$B_m :=\{x \in \Bbb R^n : ||x|| \le m\}$ and $\mathscr{F}f$
denotes the $L^2$ fourier transform of an $f \in L^2(\Bbb R^n)$.
I am trying to show that
If $f \in L^2(\Bbb R^n)$ then $f(y)=\lim\limits_{...
CommunityBot
- 1
1
vote
1
answer
113
views
The Fourier projection mappings $\{ P_N \}$ form an equicontinuous family of linear maps on $E'(S^1)$ as well?
Let $S^1=\mathbb{R}/\mathbb{Z}$ and define the Fourier projection operator $P_N$ for each $N \in \mathbb{N}$ as
\begin{equation}
P_N(f)=\sum_{n=-N}^N \langle f, e_n \rangle_{L^2} e_n
\end{equation}
...
- 407
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 ...
6
votes
1
answer
310
views
Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
- 1,075