3
votes
1answer
68 views

asymptotical behavior of integral

I'm interest in the asymptotical of $$\int_{-\pi}^{\pi}\exp\Big((\cos z+i\alpha\sin z-1)t\Big)dz\hspace{3mm}\text{as}\hspace{2mm}t\to\infty$$ for $-1<\alpha<1$. Numberical result suggest that ...
0
votes
0answers
40 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
2
votes
0answers
24 views

Asymptotic behaviour of oscillating integral

I'm interested in the big $x$ ($x \to \infty$) behaviour for the following integral $\int_{-\infty}^{\infty} \frac{dk}{\sqrt{k^2+1}} \frac{e^{-\sqrt{k^2+1}/2}}{1-e^{-\sqrt{k^2+1}}} e^{ikx}$ After a ...
2
votes
1answer
66 views

Big O proof of Fourier Coefficient

Let $f(x)$ be a $2\pi$ periodic function on R. Assume that Hölder continuous: $$\sup_{x\neq y} \frac{|f(x)-f(y)|}{|x-y|^{-\alpha}} \leq C$$ for some constants $C$ and $\alpha \in \,]0,1]$. Prove ...
0
votes
1answer
53 views

Conditions for $o(|u|^{-1})$ decay of the Fourier transform of a bounded variation function

As the question suggests I am looking for a (not very restrictive) condition on a function of bounded variation so that its Fourier transform is $o(|u|^{-1})$ as $|u| \to \infty$. Let me elaborate on ...
4
votes
1answer
194 views

How to analyze the asymptotic behaviour of this integral function?

Based on the asymptotic analysis of correlation functions at large distence in Physics, now I get a math question. Let the function $$f(x)=\int_{-1}^{1}\sqrt{1-k^2}e^{ikx}dk.$$ Without working out ...
10
votes
4answers
361 views

Singular asymptotics of Gaussian integrals with periodic perturbations

At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$, $$ \int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
1
vote
0answers
48 views

What is the relationship between singularities for complex times and high frequency asymptotics?

As said in a paper I am reading on p 2677 in the text directly above FIG3, this should be a standard result about Fourier transforms of analytic functions. In the paper the authors use these methodes ...
11
votes
3answers
363 views

The boundedness of an integral

Is there a constant $C$ which is independent of real numbers $a,b,N$, such that $$\left| {\int_{-N}^N \dfrac{e^{i(ax^2+bx)}-1}{x}dx} \right| \le C?$$
8
votes
2answers
1k views

Compactly supported function whose Fourier transform decays exponentially?

It's well known now that a function can not be compactly supported both on the space side and the frequency side (so-called uncertainty principle). On the other hand a function can have exponential ...
2
votes
1answer
115 views

Asymptotics of an improper integral

I have to show that if $x \to \infty$, then $$ \int\limits_{\mathbb{R}^d} \frac{e^{i\xi x}}{\xi^2 + 2k\xi}d\xi = O\left(|x|^{-\frac{d-1}{2}} \right) \;\;\; \; d\geqslant2, \;\;\; k\in \mathbb{C}^d ...
2
votes
1answer
276 views

Asymptotic error of Fourier series partial sum of sawtooth function

In Iwaniec's book, Topics in Classical Automorphic Forms, pg. 4, he gives the statement: $$\{x\}=\frac{1}{2}-\sum_{n=1}^N\frac{\sin 2\pi nx}{\pi n}+O((1+||x||N)^{-1})$$ where $\{x\}$ denotes the ...