A class function on a group is a function constant on every conjugacy class. Equivalently, for all group elements .
Normalized Haar measure makes this averaging a norm-one projection onto continuous class functions. Averaging a trace coefficient replaces its coefficient endomorphism by . On an irreducible summand this average is by Schur lemma. It follows that averaging finite-dimensional matrix coefficients yields finite linear combinations of irreducible characters. Together with the Peter-Weyl theorem, this proves uniform density of those characters among continuous class functions.
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In object-oriented programming, a class function (also known as a class method) is a method that is associated with a class rather than with instances of the class. This concept is most commonly found in languages like Python, Java, and C++, where you can define methods that act on the class itself rather than on individual objects. ### Key Characteristics of Class Functions: 1. **Binding to Class**: Class functions are called on the class itself rather than an instance of the class.