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Results tagged with ca.classical-analysis-and-odes
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user 1004
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
23
votes
Accepted
A question of Erdős on equidistribution
The statement was shown to be false by J. Bourgain in a paper published in 1988 (Almost Sure Convergence and Bounded Entropy, doi:10.1007/BF02765022). Well before either Harman's 1997 book and the 200 …
- 17.1k
21
votes
0
answers
1k
views
Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$
A while ago, I came across the following problem, which I was not able to resolve one way or the other.
Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t …
- 17.1k
16
votes
Accepted
On the average of continuous functions $f:\mathbb{R}^2\rightarrow[0,1]$
Yes, any such $f$ is constant. In fact, if we relax the condition so that $f$ is only required to be bounded below, but not above, then it is still true that $f$ is constant. This can be proven by mar …
- 17.1k
15
votes
Accepted
Real analytic function, injective, non surjective and preserving the rationals ?
The statement in the question is not true. Given any two enumerable and dense sets in open intervals of the reals, there is a (complex) analytic1 function giving a bijection between them. See the foll …
- 17.1k
14
votes
What is the standard notation for a multiplicative integral?
As I mentioned in my comment, this concept is the time-ordered exponential (as I remember from Quantum Field Theory lectures, long ago). Alternatively, the path-ordered exponential.
I'm no expert here …
- 17.1k
13
votes
Accepted
Have commuting functions a common value ?
Yes. The set of fixed points of $g$ is closed, nonempty, and is mapped into itself by $f$. Letting $a\le b$ be, respectively, the minimum and maximum fixed points of $g$, we have $f(a)\ge a=g(a)$ and …
- 17.1k
9
votes
Accepted
Consequence of equidistribution or not?
No, you cannot put any better bound than SN = o(N). There is a general technique, using the Baire category theorem of proving the existence of counterexamples to problems like this (which I discovered …
- 17.1k
6
votes
Accepted
Continuous dependence of the expectation of a r.v. on the probability measure
This problem reduces quickly to Holder continuity of the operator square root. That is, there exists a $C > 0$ such that
$$
\begin{align}
\lVert\sqrt{A}-\sqrt{B}\rVert\le C\lVert A-B\rVert^{1/2}&&{\rm …
- 17.1k
3
votes
Accepted
Limit connected with a periodic function
No, convergence does not hold for all irrational $x$. I posted a full answer to the question at math.stackexchange and I'll summarize the result here.
There are uncountably many values of $x$ for whi …
- 17.1k
2
votes
Hilbert transforms of measures
Showing that the two definitions agree almost everywhere is easy! Using the truncated transform
$$
\mathcal{H}\_\epsilon\mu(x)=\frac1\pi\int_{\lvert y-x\rvert > \epsilon}\frac{d\mu(y)}{x-y}
$$
then, b …
- 17.1k
1
vote
Euclidean volume of the unit ball of matrices under the matrix norm
I had a go at this question, but the method I tried here doesn't quite work out. It does reduce it to upper triangular matrices, although that doesn't seem to be a lot of help for general n.
Let your …
- 17.1k