From ca43e5660ccbd3f4beaaa2073ce7dda05a80eff6 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Mon, 1 Aug 2022 19:58:09 +0200 Subject: Kugel: Introduction and preliminaries (not spherical harmonics, yet) --- buch/papers/kugel/introduction.tex | 35 +++++++++++++++++++++++++++++++++++ 1 file changed, 35 insertions(+) create mode 100644 buch/papers/kugel/introduction.tex (limited to 'buch/papers/kugel/introduction.tex') diff --git a/buch/papers/kugel/introduction.tex b/buch/papers/kugel/introduction.tex new file mode 100644 index 0000000..5b09e9c --- /dev/null +++ b/buch/papers/kugel/introduction.tex @@ -0,0 +1,35 @@ +% vim:ts=2 sw=2 et spell tw=78: + +\section{Introduction} + +This chapter of the book is devoted to the sef of functions called +\emph{spherical harmonics}. However, before we dive into the topic, we want to +make a few preliminary remarks to avoid ``upsetting'' a certain type of +reader. Specifically, we would like to specify that the authors of this +chapter not mathematicians but engineers, and therefore the text will not be +always complete with sound proofs after every claim. Instead we will go +through the topic in a more intuitive way including rigorous proofs only if +they are enlightening or when they are very short. Where no proofs are given +we will try to give an intuition for why it is true. + +That being said, when talking about spherical harmonics one could start by +describing their name. The latter may be a cause of some confusion because of +the misleading translations in other languages. In German the name for this +set of functions is ``Kugelfunktionen'', which puts the emphasis only on the +spherical context, whereas the English name ``spherical harmonics'' also +contains the \emph{harmonic} part hinting at Fourier theories and harmonic +analysis in general. + +The structure of this chapter is organized in the following way. First, we +will quickly go through some fundamental linear algebra and Fourier theory to +refresh a few important concepts. In principle, we could have written the +whole thing starting from a much more abstract level without much preparation, +but then we would have lost some of the beauty that comes from the +appreciation of the power of some surprisingly simple ideas. Then once the +basics are done, we can explore the main topic of spherical harmonics which as +we will see arises from the eigenfunctions of the Laplacian operator in +spherical coordinates. Finally, after studying what we think are the most +beautiful and interesting properties of the spherical harmonics, to conclude +this journey we will present a few real-world applications, which are of +course most of interest for engineers. + -- cgit v1.2.1