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Results tagged with harmonic-analysis
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user 116555
Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.
3
votes
1
answer
148
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A question on a simple integral with a singular kernel?
I asked this question on math.stackexchange:
Does this integral converge when $\frac{1}{p}+\frac{1}{q}\ge1$?
No answers or very useful comments there.
May be it is more appropraite for mathoverflow.
F …
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4
votes
1
answer
209
views
estimate a singular integral using a dyadic decomposition
Let $0<\alpha_{j}<1$, $j=1,\dots,d+1$. I am trying to estimate the following singular integral:
$$I(y_{1},\dots,y_{d},z):=\int_{\substack{ x\in[0,1]^{d}\\
1/2<|x|<1}}
\frac{d x_{1} \dots d x_{d}}{|x_{ …
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5
votes
3
answers
309
views
The integrability of $\widehat{e^{-|x|^a}}$, $a>0$
Let $f_a(x)=e^{-|x|^a}$, $x\in \mathbb{R}^n$. Then $f \in L^p(\mathbb{R}^n)$ for every $1\leq p\leq \infty$. It is also smooth away from the origin and decays faster than any polynomial as $|x|\righta …
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1
vote
The integrability of $\widehat{e^{-|x|^a}}$, $a>0$
I would like to thank @Lorenzo Pompili and @Christian Remling for their insightful answers.
I found a straight-forward proof in Lemma 2.1 in
"C. Miao, B. Yuan, B. Zhang, Well-posedness of the Cauchy p …
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3
votes
1
answer
246
views
A sharp estimate for an oscillatory integral with a simple phase
Let $\alpha>1$ not necessarily an integer, and let $\psi:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function with compact support. Consider the oscillatory integral $$I(\lambda):=\int_{0}^{\infty}\ …
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0
votes
1
answer
148
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The asymptotic behaviour of a singular integral
Given $0<\alpha, \beta<1$, $a,b>0$, $a^2+b^2<1$.
I am trying to determine the asymptotic behaviour of
$$F(a,b):=\int_{\substack{a/2<x<2a\\\\b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b}}\frac{dx}{|x-a|^{\alpha}| …
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5
votes
1
answer
299
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Maximal operator estimates for the Schrödinger equation
Let $a>0$ and consider the operator
$$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$
When $a=2$, the function $Tf$ solves the Cauchy problem as …
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0
votes
1
answer
613
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Does this dyadic sum converge?
Let $a\in (0,1)$ and define
$$J(j):=\int_{0}^{\infty} e^{- 2^{j} s} \frac{s^{a}}{1+s^{2a}} ds,\quad j\in \mathbb{Z}.$$
Note that rescaling $2^{j} s\mapsto s$ shows that
$$J(j)\leq 2^{-j(1+a)}\int_{0} …
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0
votes
0
answers
92
views
The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric f...
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
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \infty, …
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5
votes
0
answers
242
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Is there a way to solve this integral on the sphere explicitly?
Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that
$k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral
$$f(y):=\int_{\mathbb{S …
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