Questions tagged [oscillatory-integral]
For questions about definite integrals $\int_a^b f(x)\,dx$ where the integrand is oscillating, often of the form $f(x) =g(x) e^{i h(x)}$.
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Asymptotic of integral involving a theta function [closed]
I would appreciate some help with theta functions. Consider $\theta_3(u,q)$:
$$
\theta_3(u,q) = 1 + 2 \sum_{n = 1}^{+\infty} q^{n^2} \cos(2 n u)
$$
I am interested in the asymptotic of the following ...
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2
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1
answer
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Intuitively, why does $I(\lambda)$ decay as $\lambda \to \infty$ if $\Phi$ is not constant?
I'm quoting a few lines from Sogge's Fourier Integrals in Classical Analysis.
Stationary phase is of central importance in classical analysis since integrals of the form
\begin{equation}
I(\lambda) = ...
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votes
1
answer
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The asymptotic of an integral $I$
Consider the integral
$$
I(\lambda)=\int_0^1 \frac{1}{\sqrt{v}}\,\left( \int_{-\infty}^{+\infty} \frac{e^{i\lambda u (u^2-v)}}{\sqrt{u^2+ 4v}}\,\varphi(u,v)\, du\right) dv,
$$
where $\varphi\in C_0^\...
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4
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The oscillatory integral $\sup_{b, z > 0} \left| \int_0^b \frac{\cos(z \sqrt{b + x}) - \cos(z \sqrt{b - x})}{x} dx \right| < \infty$
I am trying to prove the boundedness of certain oscillatory integrals and I can not deal with the following situation, which I have reduced to a specific example. I claim that
$$
\sup_{b, z > 0} \...
- 121
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0
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Integrating products of many oscillating functions
I'm working with the circular random matrix ensembles, in particular the circular unitary ensemble (CUE). For unitaries drawn from the CUE with dimension $N$, the distribution of its eigenvalues is ...
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vote
1
answer
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Oscillatory integral and Riemann integral
Consider the summation, with parameter $a \ge 0$ and non-negative integer $M = 1, 2, 3...$
$$S (a, M) = \sum_{m=1}^M \frac{2}{1+M} \sin\left( \frac{\pi m}{M+1}\right) \sin\left( \frac{\pi m M}{M+1} \...
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Characterisation of what can be written as an oscillatory integral?
Let $\chi$ be a $C_c^\infty$ function which equals $1$ on $|x|<1.$ Define
$$
I_{\Phi, \epsilon} = \int e^{i\Phi(x,\theta)}a(x, \theta) \chi(\epsilon \theta) d \theta
$$
for a phase function $\Phi$ ...
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Next term in the Stationary Phase Lemma expansion in dimension 2
Consider two functions $f\in\mathcal{S}(\mathbb{R}^2)$ and $\phi\in C^\infty(\mathbb{R}^2)$ satisfying that $\nabla\phi(x_0,y_0)=0$ and $\mathrm{det}\text{ }\mathrm{Hess}\text{ }\phi(x_0,y_0)\neq 0$, ...
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Applications of highly oscillatory integrals
I was reading a series of articles on numerical integration of highly oscillatory functions, e.g.,
S. Olver, Numerical approximation of highly oscillatory integrals
S. Xiang, H. Wang, Fast ...
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2
votes
1
answer
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$L^1$ norm of spherical/circular Dirichlet kernel
I'm currently studying a particular Fourier multiplier and I came across the following question.
The cubic $d$-dimensional Dirichlet kernel is
\begin{equation}
D_n(x)=\prod_{i=1}^d D_n^1(x_i),
\end{...
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0
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Oscillatory Integrals near the Riemann singularity
The question comes from E. M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, we concerned about the highly oscillatory distribution
$$
D(x)=\mathrm{p.v.} \...
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1
vote
1
answer
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Numerically compute an oscillating series
I would like to compute in a numerically stable way an oscillating series. Imagine I have a signal $C(n)$, $n\in\mathbb{N}$ which decays exponentially with $n$ e.g. $C(n) = e^{-2n}$. Also, imagine I ...
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Gronwall lemma with highly oscillatory kernel
As a toy model for a larger problem, I want to show that if $A,B\geq 0$ and $u(t)$ satisfies
$$|u(t)|\leq A+\left|\int_0^t B\cos(s^2)u(s)ds\right|$$
then $u$ satisfies a bound like
$$|u(t)|\leq AC$$
...
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Resource recommendation: multidimensional stationary phase method on polygons
Consider the oscillatory integral of the form
$$I(\lambda) = \int_D a(\mathbf x) e^{i\lambda \mathbf x^T A \mathbf x} d\mathbf x,$$
where $D\subset \mathbb R^n$ is a box (or more generally a polygon), ...
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Dirac delta doublet function in simple harmonic oscillation. Conditions imposed?
I'm considering an at-rest simple harmonic oscillator (m,k) and want to model the force by a doublet (derivative of dirac delta) at t=0.
$$f = \delta'(t)$$
I've already considered the case for a dirac ...
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