Math konspekt

some mathematics

[mathkonspekt]

Approximate differentiability

Here is an interesting question on mathoverflow about approximate differentiability of a function. I use to think those two definitions are equivalent.

Definition: Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. We say that $f$ is approximately differentiable at $x\in E$ if there is a linear function $L:\mathbb{R}^n\to\mathbb{R}$ such that for any $\varepsilon>0$ the set $$ \{ y\in E:\, \frac{|f(y)-f(x)-L(y-x)|}{|y-x|} <\varepsilon \} $$ has $x$ as a density point.

Then this is the same as

A measurable function $f:E\to\mathbb{R}$ defined in a measurable set $E\subset\mathbb{R}^n$ is approximately differentiable at $x\in E$ if and only if there is a measurable set $E_x\subset E$ and a linear function $L:\mathbb{R}^n\to\mathbb{R}$ such that $x$ is a density point of $E_x$ and $$ \lim_{E_x\ni y\to x} \frac{|f(y)-f(x)-L(y-x)|}{|y-x|} = 0. $$

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My favorite series is

$$\sum_{n=1}^{\infty} \frac{1}{2^n}.$$

And this is return post to Tumblr. I am not quite sure about writing here.

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Lorentz spaces

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(via https://www.youtube.com/watch?v=wTam3Krt3ec)

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Complex analysis short notes #6

Here one of my favorite examples http://math.stackexchange.com/a/484118/23566

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Complex analysis short notes #5

Also look at this example http://math.stackexchange.com/q/608023/23566.

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Complex analysis short notes #4

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Complex analysis short notes #3

Power series --- one of the main tools in complex analysis.

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Complex analysis short notes #2

The Riemann zeta function appears here.

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Complex analysis short notes #1

Complex numbers first.

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I absolutely love writing with a fountain pen. A couple of years ago my cousin from Germany gave me one of those Lamy fountain pens (I believe it is Lamy Al-star). Every time I take this pen I feel inspiration and enthusiasms to write.

However the true fan of fountain pens is the author of that post.

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Lorentz space

What are Lorentz spaces $L_{pq}$? Shortly speaking, they are generalizations of Lebesgue spaces $L_p$, and were introduced by George Lorentz in 1950.

Here I will collect books and sources which would help to learn about Lorentz spaces (no doubt the wikipedia is the first step). 1. Introduction to Fourier analysis on Euclidean spaces by Elias Stein & Guido Weiss. See chapter V section 3, where authors start with distribution function and nonincreasing rearrangement and then define $L_{pq}$ space. 2. Terence Tao notes for 247A is also good starting point to learn Lorentz spaces. Also 247A Notes on Lorentz spaces.

3. Another big book is Classical Fourier Analysis by Loukas Grafakos. 4. Introduction to Lorentz spaces (which I believe is a diploma paper) seems to contain comprehensive introduction from functional analysis point of view. Another master thesis on the topic  Decreasing  Rearrangement  and Lorentz $L(p, q)$ Spaces by Erik Kristiansson.

to be continued...

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(via https://www.youtube.com/watch?v=rUkAVQWku6s)

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When $n=1$ all harmonic functions are affine.

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Towards stochastic integral

$(\Omega, \mathcal F, {\mathcal F_t }_{t\geq0}, \mathbb P)$ with usual conditions. Define $H_2 = {X(t):[0,\infty]\to\mathbb R : X(t)\sim \mathcal F_t \text{ and } \int_0^T X^2(t)\, dt < \infty}$

Lemma. Let $X(t)\in H_2$ be a bounded continuous stochastic process. Then there is sequence of piecewise-constant processes ${X_n(t)}$ such that $$E\int_0^T(X(t)-X_n(t))^2\,dt \to 0 \text{ as } n\to\infty.$$

Proof.

Let ${0=t^n_0<\dots<x^n_n=T}$ be a sequence of decompositions of the interval $[0,T]$ s.t. $\max (t_i-t_{i-1}) \to 0$.

Define $$X_n(t) = X(t_j) \text{ for } t\in[t^n_j,t^n_{j+1}).$$ Then $X_n(t)$ is a bounded step-function and $X_n(t)\sim\mathcal F_{t_j}\subset\mathcal F_t$.

Due to continuity $$X_n(t) \to X(t) \text{ as } n\to\infty \text{ almost surely }. $$

By the dominated convergence theorem we conclude$$\int_0^T(X_n(t) - X(t))^2\, dt \to 0 \text{ as } n\to \infty \text{ almost surely }.$$

Then using the same theorem we achieve $$E\int_0^T(X(t)-X_n(t))^2\,dt \to 0 \text{ as } n\to\infty.$$

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This is the second one!