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Results tagged with ca.classical-analysis-and-odes
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user 4281
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
7
votes
Freshman's definition of sin(x)?
When I was first taught analysis, I do remember things being in a slightly strange order. What annoyed me most was how we were expected to do an exercise involving say, the sine function before we had …
- 1,771
2
votes
Proving theorems by using functions with fixed points.
I am familiar with a good example from the theory of 2nd order elliptic PDE. Technicalities omitted...
A special case of the Leray-Schauder Theorem says the following:
Let $T$ be a compact mapping o …
8
votes
Applications of Rademacher's Theorem
The application I am most familiar with is that it is used in the proof of the following result:
Suppose $f : \mathbb{R}^n \to \mathbb{R}$ is Lipschitz. For any $\epsilon > 0$, there exists a $C^1$ f …
- 1,771
6
votes
Accepted
Is the derivative of a Lipschitz function better than L^\infty
Lipschitz functions are exactly $W^{1,\infty}$ (See 'Sobolev space' on wikipedia - under other examples and perhaps the bit about absolute continuity on lines). This means the short answer to your que …
- 1,771
21
votes
3
answers
6k
views
Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure th...
What is the point of $\pi$-systems and
$\mathcal{D}$ / Dynkin /
$\lambda$-systems?
I am an analyst in the process of consolidating my measure theory knowledge before moving on to harder/newe …
- 1,771
4
votes
2
answers
2k
views
Inclusions of $C^{k,\alpha}$ spaces
When is $C^{k,\alpha}(\bar{\Omega})$ a
subset of
$C^{k',\alpha'}(\bar{\Omega})$?
Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k …
- 1,771
7
votes
4
answers
6k
views
The characteristic (indicator) function of a set is not in the Sobolev space H¹
Is it true that the characteristic
(indicator) function of a subset of
Euclidean space with finite positive
measure is never in the Sobolev space
$H^1 = W^{1,2}$? And if so, what is the best/easiest/ …
- 1,771
12
votes
Taking "Zooming in on a point of a graph" seriously
As Rbega says in the comments, if you are really keen to see this rescaling idea put to use in a more rigorous or advanced way, then you can look at some Geometric Measure Theory. While it will look v …
- 1,771
5
votes
1
answer
135
views
Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to a...
Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?:
Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that the …
- 1,771
11
votes
Learning roadmap for harmonic analysis
To my mind, the classical subject is quite different from the modern, evolved form of the subject
I started on the classical side with Yitzhak Katznelson's An Introduction to Harmonic Analysis: This …
13
votes
Why should one still teach Riemann integration?
Although not a direct answer to the question, this may be relevant to the discussion:
I learnt basic measure theory and the theory of Lebesgue integration in a course called "Probability and Measure" …