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17 questions
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2 votes
1 answer
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Consider the following complex oscillatory integral on a neighborhood of zero $U\subset \mathbb{R}^n$: \begin{equation} \mathscr{I}(a)=\int_{U}e^{a S(x)}\varphi(x)\, d^nx \end{equation} The functions $...
color's user avatar
  • 159
3 votes
1 answer
136 views

Let $V(x)$ be a smooth function with compact support in $[-1,1]$, say, and that is nonzero at $x=0$. In the case I'm considering, I have in mind a smooth bump function, constructed in the manner ...
Joshua Stucky's user avatar
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3 votes
0 answers
160 views

A function $f : \mathbb R \rightarrow \mathbb R$ is said to be Gevrey of order $s > 0$ if it is smooth and satisfy the estimate $$ \exists A,B > 0,\quad \forall k \in \mathbb N,\quad \forall x \...
blamethelag's user avatar
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1 vote
0 answers
111 views

Let $a\in (0,1)$, $\epsilon>0$ and $\lambda>>1$. Consider the oscillatory integral $$I(\lambda):=\int_{x>\lambda} \frac{1}{(x-\lambda)^a}\frac{\sin{(\epsilon x)}}{x}e^{i x}dx.$$ I am ...
Medo's user avatar
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2 votes
0 answers
102 views

Let $R > 0 $ and set $h = \frac{1}{R}$. Let $G \in C^\infty(\mathbb{R})\cap L^\infty(\mathbb{R})$. Further restrictions on $G$ are allowed. Consider the (R-dependent) integral operator $K_R: L^2(\...
Qualearn's user avatar
  • 143
2 votes
1 answer
454 views

For $a > 0$ and $n \in \mathbb Z_+$, consider the oscillatory integral $$\int_{0}^1 f(x) f(ax) \dots f(a^n x) \, dx,$$ where $f$ is an integrable function on $[0, 1]$, which we extend by ...
Nate River's user avatar
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1 vote
0 answers
113 views

Suppose I am interested in computing $$ I \equiv \int_0^B dx \, g(x) f(x) $$ where $B$ is a known upper bound for the integral, $g(x)$ is a known oscillating function and $f(x)$ is a smooth function ...
knuth's user avatar
  • 33
3 votes
0 answers
181 views

Let $\lambda>1$, $x,y\in [0,1]$, and $K_\lambda(x,y)=(1+\lambda|x-y|)^{-2}e^{i\lambda|x-y|}$. Consider the operator $$T_\lambda f(x)=\int_0^1 K_\lambda(x,y)f(y)dy.$$ We want to precisely estimate ...
capitalone's user avatar
  • 31
2 votes
1 answer
277 views

Consider the following integral for $f \in C_c^{\infty}(\mathbb R^n)$, $x_0$ fixed (possibly zero), and $n \ge 3$ $$F(\lambda) = \int_{\mathbb R^n} e^{i\lambda \vert x-x_0 \vert^2} \frac{f(x)}{\vert x ...
António Borges Santos's user avatar
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1 vote
0 answers
233 views

Let $\phi$ be a smooth real function in one variable and say $w$ is a smooth function with compact support say $[- 1, 1]$. Let me define $$ I_{\lambda} = \int_{\mathbb{R}} w(t) e^{i \lambda \phi(t)} ...
Johnny T.'s user avatar
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4 votes
2 answers
504 views

Question: Given exponents $0<\alpha<\beta$ and an interval $[a,b]\subset(0,\infty)$ are there constants $C,d>0$ such that for any $\lambda_1,\lambda_2\in\mathbb{R}$, $$\left|\int_a^be(\...
Joel Moreira's user avatar
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3 votes
0 answers
415 views

The paper in question is A Remark on Schrodinger Operators. The goal of the argument is to estimate the following integral: $$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
Dispersion's user avatar
  • 946
1 vote
1 answer
232 views

I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator $...
Medo's user avatar
  • 900
4 votes
1 answer
272 views

I posted this on Stackexchange but got no responses or comments. Consider the following integral, for $\epsilon\ne 0:$ $$\displaystyle\frac{1}{(2\pi)^2\epsilon^4}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[...
Josh Lackman's user avatar
  • 1,217
5 votes
1 answer
276 views

I am having a little confusion in verifying the two dimensional oscillatory integral in Lemma 2.1 in This paper, namely $$I_t (x,y) = \int_{\mathbb{R}^2} |\xi|^{\epsilon + i \beta} e^{i t(\xi^3 + \xi \...
Mr. Proof's user avatar
  • 169

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