Context in Harmonic Analysis
The theory of unitary representations of groups is closely connected with harmonic analysis. In the case of an abelian group G, a fairly complete picture of the representation theory of G is given by Pontryagin duality. In general, the unitary equivalence classes of irreducible unitary representations of G make up its unitary dual. This set can be identified with the spectrum of the C*-algebra associated to G by the group C*-algebra construction. This is a topological space.
The general form of the Plancherel theorem tries to describe the regular representation of G on L2(G) by means of a measure on the unitary dual. For G abelian this is given by the Pontryagin duality theory. For G compact, this is done by the Peter-Weyl theorem; in that case the unitary dual is a discrete space, and the measure attaches an atom to each point of mass equal to its degree.
Read more about this topic: Unitary Representation
Famous quotes containing the words context, harmonic and/or analysis:
“Parents are led to believe that they must be consistent, that is, always respond to the same issue the same way. Consistency is good up to a point but your child also needs to understand context and subtlety . . . much of adult life is governed by context: what is appropriate in one setting is not appropriate in another; the way something is said may be more important than what is said. . . .”
—Stanley I. Greenspan (20th century)
“For decades child development experts have erroneously directed parents to sing with one voice, a unison chorus of values, politics, disciplinary and loving styles. But duets have greater harmonic possibilities and are more interesting to listen to, so long as cacophony or dissonance remains at acceptable levels.”
—Kyle D. Pruett (20th century)
“Analysis as an instrument of enlightenment and civilization is good, in so far as it shatters absurd convictions, acts as a solvent upon natural prejudices, and undermines authority; good, in other words, in that it sets free, refines, humanizes, makes slaves ripe for freedom. But it is bad, very bad, in so far as it stands in the way of action, cannot shape the vital forces, maims life at its roots. Analysis can be a very unappetizing affair, as much so as death.”
—Thomas Mann (1875–1955)