Approximation Theorem

Hello and welcome to my complete video course about Approximation Theorem consisting of 4 videos. This is a supplementary video to the Functional Analysis series. It connects the topic of $C^\infty$-functions to the uniform convergence of sequence of functions.

Part 1 - An Approximation Theorem for Continuous Functions

This is a classical approximation result that makes the study of continuous functions much easier.

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Content of the video:

00:00 Intro
00:55 Standard mollifier
03:21 Dirac sequence
04:12 Properties of a Dirac sequence
06:17 Formulation of the approximation theorem
08:00 Proof of the approximation theorem
13:35 Credits

Part 2 - Approximation of Integrable Functions with Test Functions

We can extend the approximations results we have already discussed. In the end, we can show that the compactly supported smooth functions lie dense in the space of integrable functions. In short $C_c^\infty(\mathbb{R}^n)$ is dense in $L^1(\mathbb{R}^n)$ with respect to the $ \lVert \cdot \rVert_1 $-norm.

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Part 3 - Testing Locally Integrable Functions

In this video we will also use the standard mollifier to smoothen a continuous function. Therefore, it fits perfectly into this small video series about approximation theorems. Moreover, we will define so-called locally integrable function on an open set $\Omega \subseteq \mathbb{R}^n$. Simply said these are function where the integral over any compact set exists. They play a crucial role in distribution theory.

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Part 4 - Approximation of Lᵖ-Functions with Test Functions

The result from part 2 above can also be generalized to every $L^p$-space. We don’t have to change too much in the proof such that we can just redo the whole video again. Check out the PDF version for the correct formulations.