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Results tagged with integration
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user 62673
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
2
votes
2
answers
391
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
3
votes
1
answer
316
views
Bounding a series of nested integrals
Consider the following matrix function
$$
f(t) = \cos(\omega_1t) A_1 + \cos(\omega_2t) A_2, \quad t\ge 0,
$$
where $A_1$, $A_2$ are real square matrices and $\omega_1$, $\omega_2$ positive numbers.
N …
- 2,712
1
vote
0
answers
187
views
A question concerning positive definite matrix functions
Let $C(e^{i\theta})$ be an $m\times n$ ($m\ge n$) matrix-valued continuous function of $\theta\in[-\pi,\pi]$. Let $A_1(e^{i\theta})$ and $A_2(e^{i\theta})$ be two $n\times n$ positive definite matrix- …
- 2,712
4
votes
1
answer
406
views
On a vanishing integral inner product
Let $G(z)$ be an $n\times m$ rational matrix-valued function of full column rank on the unit circle. Further, let $P(z)$ be an $m\times m$ rational matrix-valued function positive definite on the unit …
- 2,712
1
vote
On a vanishing integral inner product
I think I've managed to provide an answer to the second (simpler) question. I'll post it here.
Let $P(e^{i\theta})=Mp(e^{i\theta})$. (In what follows, I will drop the dependence on $\theta$ to lighte …
- 2,712