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Results tagged with ca.classical-analysis-and-odes
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user 9833
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
4
votes
Analytic approximations of smooth vector fields
I believe the most natural approach to this particular question is via Fourier analysis. In the periodic case we have the series
$$u(x)=\sum_{k\in\mathbb{Z}^3}u_k e^{2\pi i (k,x)},$$
and the condition …
- 6,891
6
votes
Lindelöf hypothesis claim
(Not an answer of any sort, just too long for a comment.) The main result
seems to be an integral equation (1.3) of the form
$$\int_{-\infty}^\infty K(t,\tau) |\zeta(\tfrac{1}{2}+it\tau)|^2\,d\tau= …
- 6,891
2
votes
Closed-Form solution for system of simple nonlinear equations
(Not really an answer, but hopefully may be helpful.)
It is convenient to consider this as a problem of finding a solution of the equation
$$(N-2)x-\sum_i\delta_i\sqrt{x^2-d_i}=0,$$
for a given vect …
- 6,891
1
vote
asymptotic with a very degenerate stationary phase
(I am not really an expert, so it should be taken with a pinch of salt.)
One may take $f=1$ and simply compute the integral using polar coordinates. (There are minor problems with convergence, but …
- 6,891
3
votes
Are there any techniques for solving a differential equation of the form $f ' (x) = f( f( x ...
For what I know, the standard method is the Taylor series expansion
at a fixed point, i.e. at a point $x=a$ such that $f(a)=a$.
- 6,891
2
votes
Complex structure on $L^2(\mathbb R)$ generalizing the Hilbert transform
EDIT: This solution does not satisfy the third condition,
which rules out the Hilbert transform itself. So, this is an answer to
different question. I do not delete it in hope it may be useful for so …
- 6,891