Harmonic analysis

The ( abstract ) or harmonic analysis ( abstract ) harmonic analysis is the theory of locally compact groups and their representations. The name comes from the fact that there is an analog to the Lebesgue measure on the real numbers measure on any locally compact groups, the so-called hair - measure. With respect to this measure can be - depending on the additional properties of the group, particularly in commutative groups - the theory of Fourier Analysis transferred. This leads to important insights on locally compact groups. This article focuses on the representation of the generalizations of the classical situation in the real numbers.

• 4.1 Compact Groups
• 4.2 Plancherel measure of unimodular groups
• 4.3 Non - unimodular groups

Locally Compact Groups

A locally compact group is a topological group which carries a locally compact topology. Example of this are:

• The real numbers with addition as group join form with the Lebesgue measure as a hair -dimension prototype theory.
• Of the addition and the n-dimensional Lebesgue measure is a simple generalization of the first embodiment.
• Each group with the discrete topology is locally compact. The hair - measure is the counting measure.
• The circle is a compact group with multiplication as group link. The Haar measure is the size of the image, said on [0,1 ] is given by the Lebesgue measure. This group also plays an important role in the further course.
• The group of invertible matrices with the matrix multiplication is an example of a non-commutative locally compact group. The type of hair - measure requires advanced knowledge integration. Is this Lebsgue measure on the so is given by a hair - measure. In general, non-commutative case, one has to distinguish between left - and right - Hair measure, in this example is not the necessary.

The Banach algebra L1 ( G)

Is the hair - measure on the locally compact abelian group G, one can regard this measure the space L1 ( G ) form. It is the Banach space of complex-valued L1 - functions, where almost everywhere matching functions are identified in the usual way. As in the case of real numbers defining the folding

Multiplication, which makes it a commutative Banach algebra. The shortcut on G is written additively, is to be calculated in G! By the formula

Is defined on the Banach algebra is an isometric involution. Using similar formulas we can define a Banach algebra in the non-commutative case; which is carried out in the article groups C * algebra.

As with the group algebra of the algebraic representation theory of groups, to representations of locally compact groups in a natural way in Algebrendarstellungen of them translated and vice versa. This transition is also important for the definition of the Fourier transform.

Abelian groups

Dual group

Let be a locally compact abelian group. A steady group homomorphism is called a character of G. The set of all characters is denoted by. With the multiplication is added to a group. With the topology of compact convergence even becomes a locally compact abelian group that can therefore also referred to as dual group of G. We consider some examples:

• Each character has the shape of a. So you identify with z, one has, at least as quantities. One can show that this identification in the sense of locally compact groups is okay.
• Each character is of the form a. In this sense, so you have.
• The characters are for, which leads to duality.

The last example behaves inversely ' to the previous. This is no coincidence, because it is the following duality theorem of Pontryagin.

Duality theorem of Pontryagin

If G is a locally compact abelian group, then.

This theorem justifies the term dual group, because you can recover the starting group back out of the dual group.

Fourier transformation

Ie If G is a locally compact abelian group with hair - measure and is so

The Fourier transform of f is obtained in the case due to the classic Fourier transform. Many properties of the classical Fourier transform remain in the abstract case. Thus, for example, be on a continuous function, which disappears at infinity. The Fourier transform is an injective homomorphism.

The view of the physicist to the classical Fourier transform is that an arbitrary ' function as the sum ( = integral) can be represented by harmonic oscillations, because solves the undamped oscillation equation. This view is also retained in the abstract frame, the harmonic vibrations must - at least in the abelian case - to be replaced only by the characters. For this reason we speak of abstract harmonic analysis.

Fourier inversion formula

The Fourier inversion formula is preserved in this abstract framework. If G is locally compact group with our dual group, and is hair - measure on the dual group, one set for

Is then such that the Fourier transform is in, we obtain by means of this inversion formula of f back again, at least up to a constant factor. This constant factor stems from the fact that the hair - measure is unique only up to a constant factor. Even in the prototypical case of real numbers the well-known factor occurs when you use on the group and the dual group of the Lebesgue measure.

Fourier series

A function F on the circle group can be understood in an obvious way as a - periodic function f on, you put it. Here, the Fourier - transform of F is a function of:

We see here the Fourier coefficients of f, the Fourier inversion formula then leads to the well-known Fourier series. Thus, the abstract harmonic analysis provides the framework for a common theoretical consideration of both the classical Fourier transform and the Fourier series expansion.

Gelfand representation

Let G be a locally compact abelian group again with hair - measure. The Fourier transform can also be interpreted in the following manner. Each character is defined by the formula

A steady, linear multiplicative functional on. The Fourier transform proves to be the Gelfand transform of the commutative Banach algebra.

Non - Abelian Groups

For non- abelian groups it is not enough to consider characters of the group, instead one considers unitary representations on Hilbert spaces. So be a locally compact topological group. A unitary representation of a Hilbert space is now at a constant group homomorphism, said unitary group call provided with the weak operator topology, which in this case coincides with the strong operator topology. There is now a sub- Hilbert space of such that for all still so can display on limit, ie invariant subspace of the representation. A representation for which no non-trivial invariant subspace there is, irreducible. Now, you select a system of representatives of the irreducible representations of a group with respect to unitary equivalence. In the abelian case this corresponds precisely to the characters. Since each such representation can be continued in some canonical way to a Algebrendarstellung on by

Prepared in a suitable sense of integration, can be for a family

Define the Fourier transform is known.

Further sets of harmonic analysis are now dealing with how and when and the space that can be equipped with suitable structures that are obtained by the Fourier transformation (similar to the statement of Plancherelformel ), whereby the Fourier transformation can be reversed. However, such a result for all locally compact topological groups could not be obtained.

Compact groups

A far-reaching generalization of the Fourier transform on compact groups provides the set of Peter -Weyl. This sentence is particularly elementary, since the structure of, in a sense "discrete" ( compact in the abelian case actually discreet as topological space ) and can easily be construed as an orthogonal sum of matrices.

Plancherel measure of unimodular groups

In the event that the group is unimodular and zweitabzählbar and a certain representation theoretic property has (type - 1 group, ie the group C * - algebra is postliminal ) can be equipped with the Plancherel measure, with respect to this measure can be a direct integral of the respective areas of the Hilbert -Schmidt operators form as elements the Fourier transforms of this space can then be interpreted and transformed back.

With respect to the Plancherel measure quantities of individual items can have positive measure, they form the so-called discrete series irreducible partial representations of the regular representation of the group. This is approximately the case in compact groups, which in turn results in the set of Peter -Weyl.

Non - unimodular groups

On non- unimodular groups inverse transformation is not possible in the same manner. The remedy here is, in some cases, special semi -invariant operators that are specific, generally only densely defined and unbounded, positive, self-adjoint closed operators, which are scaled in such a way that, in turn, can be equipped with the Plancherel measure, the Fourier transforms obtain a Hilbert space structure and a back-transformation is possible. This semi- invariant operators replace ( equivariant ) constants that are needed for scaling the unimodular case, and will be called Duflo Moore operators or formal degree operator.