All Questions
8 questions
5
votes
1
answer
314
views
A simple oscillatory integral with a non-smooth phase
Let $\phi\in C_c^\infty(\mathbb{R})$ be an even function such that $\chi_{(-1/2,1/2)}\le\phi\le \chi_{(-1,1)}$, where $\chi_{(a,b)}$ stands for the indicator function of the interval $(a,b)$. For $\...
- 421
4
votes
1
answer
168
views
Uniform decay of $J'_{\nu}(x)$ for $x\gg1$
I need a uniform decay estimate for the derivative $J'_{\nu}(x)$
of the Bessel functions.
By `uniform' I mean an estimate independent of $\nu$, at least for a range of orders like $\nu\ge0$.
For $J_{\...
- 9,007
3
votes
1
answer
256
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 $$I(\lambda):=\int_{0}^{\...
- 852
2
votes
2
answers
399
views
Asymptotic decay rate of an oscillatory integral
Consider the following oscillatory integral
$$
I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac
{(1 - \cos(2x)) (1 - \cos(2y))}
{2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y.
$$
where $...
- 2,712
2
votes
0
answers
183
views
Are there any improvements on the estimate of oscillatory integral with one-side folds?
Suppose $X$ and $Z$ are open sets in $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$, respectively. Define $T_\lambda f:L^2(Z)\to L^2(X)$ by $$T_\lambda f(x)=\int e^{i\lambda\Phi(x,z)}a(x,z)f(z)dz,$$where the ...
- 171
1
vote
1
answer
480
views
Is there an asymptotic bound for this oscillatory integral?
I have an oscillatory integral:
$$ \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
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
1
answer
150
views
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}}\...
- 852