All Questions
56 questions
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}(\...
- 1,219
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
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
- 852
6
votes
1
answer
310
views
Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
- 434
0
votes
1
answer
245
views
Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e
The Riemann-Liouville integral is defined by
$$
I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t
$$
where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
- 141
2
votes
0
answers
216
views
Fourier transform of Dirac delta distribution
Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$
$$ V(...
- 73
1
vote
1
answer
203
views
Explanation of a step in a work by C. E. Kenig and A.D. Ionescu
I am studying the work
Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...
- 159
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....
- 1,879
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 $f\in L^p(\mathbb{T})$, $2\le p<\infty$, ...
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
$$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$
...
- 873
1
vote
1
answer
133
views
Integrability of fractional heat kernel
In Estimates of fractional heat kernel, it was stated that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (...
- 109
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 $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)...
user139844
2
votes
0
answers
169
views
Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
- 839
2
votes
1
answer
295
views
Non-zero, bounded, continuous, differentiable at the origin, compactly supported functions with everywhere non-negative Fourier transforms
Do there exist functions $F(x) \! : \, \mathbb R \to \mathbb R$ which are non-zero and bounded:
$$ \mathrm {Range} (F) = [l, u] \, , \quad \mathrm {where} \quad l, u \in \mathbb R \land u > l \, ; \...
- 179
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
$$...
- 501
7
votes
2
answers
824
views
Fourier series of smooth functions in infinitely many variables
Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
- 401
-1
votes
1
answer
70
views
Is this kind of interpolation correct?
Let $f=\sum f_j$ be a finite sum. Assume that
$$ \|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$
$$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$
Then can we conclude that for $2<p<\infty$
$$\|f\|_p\le C^{1-\...
- 1
6
votes
1
answer
134
views
Multi-parameter stationary phase asymptotic expansion
I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...
- 1,692
10
votes
1
answer
586
views
Nonlinear Schrödinger equation with discrete Laplacian
In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
user127411
1
vote
0
answers
324
views
Conditions for Poisson summation (for discontinuous functions)
Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
- 3,799
2
votes
0
answers
293
views
Average of irrational flow on the torus
Let $$F(x,y) = \frac{1}{\sqrt{2-\sin(2\pi x) - \sin(2\pi y)}}$$
defined on $\mathbb{T}^2$. Here $\mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ is the 2-torus. How can I show that
$$ \lim_{T\...
- 375
1
vote
0
answers
237
views
On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
1
vote
0
answers
146
views
Functional equation with Fourier transform
What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$:
$$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$
Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....
- 1,199
7
votes
1
answer
2k
views
Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$
I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to ...
- 1,247
3
votes
1
answer
480
views
Is there a uniform upper bound for this oscillatory integral?
I am wondering whether the following uniform upper bound holds:
$|\int_a^{2a}\frac1t\sin(N b^2t)\exp(iNbt^2)dt|\le Cab^2,$
where $0<a<b<1$, $N>N_0(a,b)\gg1$, and $C$ is a constant ...
- 187
2
votes
0
answers
185
views
Is this simple oscillatory integral operator uniformly bounded on $L^2$?
Let $\phi(t,s)$ be a real-valued function smooth away from the diagonal, and equal to 0 on the diagonal. Assume that $0\le \phi(t,s)\le |t-s|$ for $t,s\in \mathbb{R}$. Let
$$T_\lambda f(t)=\int \frac{\...
- 171
2
votes
0
answers
183
views
Are there any improvements on the estimate of oscillatory integral with one-side folds?
Suppose $X$ and $Z$ are open sets in $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$, respectively. Define $T_\lambda f:L^2(Z)\to L^2(X)$ by $$T_\lambda f(x)=\int e^{i\lambda\Phi(x,z)}a(x,z)f(z)dz,$$where the ...
- 171
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
15
votes
2
answers
680
views
Are Fourier transforms of L^p stable under diffeomorphisms?
Let $\xi$ be a compactly supported distribution on $\mathbb R^n$ and assume that its Fourier transform is in $L^p$. Let $\phi:\mathbb R^n\to\mathbb R^n$ be a diffeomorphism. Does the Fourier ...
- 2,639
0
votes
1
answer
629
views
Fourier Transform of sub-Gaussian distributions
The high level question is: Just as the Fourier transform of a Gaussian is a Gaussian, is the Fourier Transform of a sub-Gaussian also a sub-Gaussian?
Let $x \in \mathbf{R}^n$ denote some sub-...
- 445
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
226
views
Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$
The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as
$$
L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}),
$$
where
$$
L^2_k (\mathbb{R}^2; \...
- 2,834
1
vote
0
answers
133
views
Condition for boundedness in stationary phase theorem
I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.
Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and $j, ...
- 93
0
votes
2
answers
425
views
A book about almost periodic functions [closed]
Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot.
- 153
5
votes
0
answers
913
views
Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
- 103
1
vote
1
answer
480
views
Is there an asymptotic bound for this oscillatory integral?
I have an oscillatory integral:
$$ \int u(x,y) e^{i\lambda f(x,y)} dx $$
with $f(x,y)\in \mathbb{C}^{\infty}$ a complex-valued function in a neighborhood of $(0,0)$ satisfying:
$$ \text{Im} f \geq ...
- 93
27
votes
2
answers
5k
views
What can be said about the Fourier transforms of characteristic functions?
What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular,
What properties are common to ...
- 2,719
6
votes
2
answers
2k
views
What is the translation in Fourier transform for a function to have exp. decay at $x\to -\infty$
It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if ...
- 319
8
votes
2
answers
3k
views
$L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$
It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in ...
- 176
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
6
votes
1
answer
591
views
For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...
user14166
8
votes
2
answers
1k
views
What is the simplest oscillatory integral for which sharp bounds are unknown?
I have either heard or read that sharp asymptotics and bounds for oscillatory integrals of the form
$ \int e^{i \lambda \Phi(x)} \psi(x) dx \quad \lambda \to \infty $
are unknown when the critical ...
- 2,243
0
votes
0
answers
104
views
Differential equation with switched parameters and boundary conditions in integral form
Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem.
...
- 43
2
votes
1
answer
607
views
Stein's extension operator and wave front sets
Let $K\subset\mathbb{R}^d$ be a compact set with non-empty interior and Lipschitz boundary. In Section VI.3 of his book "Singular Integrals and Differentiability Properties of Functions", E. M. Stein ...
- 7,817
8
votes
0
answers
349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
59
votes
7
answers
29k
views
Learning roadmap for harmonic analysis
In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
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.
...
- 4,952
24
votes
3
answers
3k
views
Can Hölder's Inequality be strengthened for smooth functions?
Is there an $\epsilon>0$ so that for every nonnegative integrable function $f$ on the reals,
$$\frac{\| f \ast f \|_\infty \| f \ast f \|_1}{\|f \ast f \|_2^2} > 1+\epsilon?$$
Of course, we ...
- 9,756
8
votes
1
answer
2k
views
Recent progress on Bochner-Riesz conjecture
Consider the family of operators $T_\delta$, $\delta \geq 0$, defined on $\mathbb{R}^n$ by
$
\widehat{T_\delta f}(\xi) = (1-|\xi|^2)_+^\delta \widehat{f}(\xi).
$
($(1-|\xi|^2)_+^\delta$ are known as ...
- 505
7
votes
1
answer
822
views
On a decomposition of L^1(G)
[EDITED by Y. Choi - I have attempted to paraphrase the original question into something a bit terser and more precise; if this is not what the original poster intended, they should make corrections ...
- 643
7
votes
3
answers
1k
views
A Question concerning the Fourier Transform of $\mathbb{R}$
Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...
- 11.2k