Harmonic analysis

The (abstract) harmonic analysis or (abstract) harmonic analysis is the theory of locally compact groups and their representations . Therefore, the name is because it on any locally compact groups to Lebesgue measure analog to the real numbers measure is, the so-called hair-measure . With regard to this measure - depending on the additional properties of the group, especially in the case of commutative groups - the theory of Fourier analysis can be transferred. This leads to important insights into locally compact groups. This article focuses on presenting the generalizations of the classical situation in real numbers.

Local compact groups

A locally compact group is a topological group that carries a locally compact topology . Examples are:

• The real numbers with the addition as a group link form the prototype of the theory with the Lebesgue measure as a hair measure .${\ displaystyle \ mathbb {R}}$
• The one with addition and the n-dimensional Lebesgue measure is a simple generalization of the first example.${\ displaystyle {\ mathbb {R}} ^ {n}}$
• Each group with the discrete topology is locally compact. The hair measure is the counting measure .
• The circle is with the multiplication as a group linking a compact group . The Haar measure is the image measure of the figure , where the Lebesgue measure is given on [0,1]. This group plays an important role in the further course.${\ displaystyle {\ mathbb {T}} = \ {z \ in {\ mathbb {C}}; | z | = 1 \}}$${\ displaystyle [0,1] \ rightarrow {\ mathbb {T}}, \, x \ mapsto e ^ {2 \ pi ix}}$
• The group of invertible - matrices with matrix multiplication is an example of a non-commutative locally compact group. The specification of the hair measurement requires advanced integration knowledge. If the Lebesgue measure is on that , then a hair measure is given. In the general non-commutative case you have to differentiate between left and right hair measure, in this example this is not yet necessary.${\ displaystyle \ mathrm {GL} (n, {\ mathbb {R}})}$ ${\ displaystyle n \ times n}$${\ displaystyle \ lambda}$${\ displaystyle {\ mathbb {R}} ^ {n ^ {2}}}$${\ displaystyle \ textstyle \ mu (A) = \ int _ {A} {\ frac {1} {| \ det (X) | ^ {n}}} d \ lambda (X)}$

The Banach Algebra L 1 (G)

If the Haar measure is on the locally compact Abelian group G, the space L 1 (G) can be formed with respect to this measure . It is the Banach space of complex-valued L 1 functions, whereby matching functions are identified in the usual way almost everywhere. As in the case of real numbers, it defines the convolution${\ displaystyle \ lambda}$

${\ displaystyle f * g (x): = \ int _ {G} f (y) g (xy) d \ lambda (y), \, \, f, g \ in L ^ {1} (G)}$

a multiplication that makes a commutative Banach algebra . The link was written additively to G and has to be calculated in G! Through the formula ${\ displaystyle L ^ {1} (G)}$${\ displaystyle xy = x + (- y)}$

${\ displaystyle f ^ {*} (x) \, = \, {\ overline {f (-x)}}}$

an isometric involution is defined on the Banach algebra. With similar formulas one can also define a Banach algebra in the non-commutative case ; this is explained in the article Group C * Algebra . ${\ displaystyle L ^ {1} (G)}$

As with the group algebra of the algebraic representation theory of groups, representations on locally compact groups can naturally be translated into algebra representations of and vice versa. This transition is also essential for the definition of the Fourier transform. ${\ displaystyle L ^ {1} (G)}$

Abelian groups

Dual group

Let be an abelian locally compact group. A continuous group homomorphism is called a character of . The set of all characters is denoted by. With the multiplication becomes a group. With the topology of compact convergence , it even becomes a locally compact Abelian group, which is therefore also called a dual group of . We consider some examples: ${\ displaystyle G}$ ${\ displaystyle \ chi \ colon G \ rightarrow \ mathbb {T}}$${\ displaystyle G}$${\ displaystyle {\ widehat {G}}}$${\ displaystyle (\ chi \ cdot \ psi) (a): = \ chi (a) \ psi (a)}$${\ displaystyle {\ widehat {G}}}$${\ displaystyle {\ widehat {G}}}$${\ displaystyle G}$

• Each character has the shape for one . If one identifies with , then one has , at least as sets. It can be shown that this identification is also okay in terms of locally compact groups.${\ displaystyle \ chi \ colon \ mathbb {R} \ rightarrow \ mathbb {T}}$${\ displaystyle \ chi _ {z} (x) = e ^ {ixz}}$${\ displaystyle z \ in \ mathbb {R}}$${\ displaystyle \ chi _ {z}}$${\ displaystyle z}$${\ displaystyle {\ widehat {\ mathbb {R}}} \ cong {\ mathbb {R}}}$
• Each character is of the shape for one . So in that sense you have .${\ displaystyle \ chi \ colon \ mathbb {Z} \ rightarrow \ mathbb {T}}$${\ displaystyle \ chi _ {z} (n) = z ^ {n}}$${\ displaystyle z \ in \ mathbb {T}}$${\ displaystyle {\ widehat {\ mathbb {Z}}} \ cong \ mathbb {T}}$
• The characters are for what leads to duality .${\ displaystyle \ chi \ colon \ mathbb {T} \ rightarrow \ mathbb {T}}$${\ displaystyle \ chi _ {n} (z) = z ^ {n}}$${\ displaystyle n \ in \ mathbb {Z}}$${\ displaystyle {\ widehat {\ mathbb {T}}} \ cong \ mathbb {Z}}$

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

Pontryagin's duality theorem

If is a locally compact Abelian group, then . ${\ displaystyle G}$${\ displaystyle {\ widehat {\ widehat {G}}} \ cong G}$

This sentence justifies the term dual group, because one can win back the starting group from the dual group.

The Fourier transform

Is a locally compact Abelian group with hair measure and is , so is called ${\ displaystyle G}$${\ displaystyle \ lambda}$${\ displaystyle f \ in L ^ {1} (G)}$

${\ displaystyle {\ widehat {f}} \ colon {\ widehat {G}} \ rightarrow {\ mathbb {C}}, \, \, {\ widehat {f}} (\ chi) = \ int _ {G } f (x) {\ overline {\ chi (x)}} \; d \ lambda (x)}$

the Fourier transform of . In this case , the classic Fourier transformation is obtained . Many properties of the classic Fourier transformation are retained in the abstract case. So is z. B. always on a continuous function that vanishes in infinity. The Fourier transform is an injective homomorphism . ${\ displaystyle f}$${\ displaystyle G = \ mathbb {R}}$${\ displaystyle {\ widehat {\ mathbb {R}}} \ cong \ mathbb {R}}$${\ displaystyle {\ widehat {f}}}$${\ displaystyle {\ widehat {G}}}$ ${\ displaystyle L ^ {1} (G) \ rightarrow C_ {0} ({\ widehat {G}})}$

The physicist's view of the classic Fourier transformation is that any function can be represented as a sum (= integral) of harmonic oscillations , because it solves the undamped oscillation equation . This view is also retained in the abstract framework, the harmonic vibrations only have to be replaced by characters - at least in the Abelian case. This is why one speaks of abstract harmonic analysis . ${\ displaystyle \ chi _ {z} (x) = e ^ {2 \ pi ixz}}$

Inverse Fourier Formula

The Fourier inverse formula is also retained in this abstract framework. If G is our locally compact group with a dual group , and if Haar measure is on the dual group, then we set for${\ displaystyle {\ widehat {G}}}$${\ displaystyle {\ widehat {\ lambda}}}$${\ displaystyle g \ in L ^ {1} ({\ widehat {G}})}$

${\ displaystyle {\ check {g}} \ colon G \ rightarrow {\ mathbb {C}}, \, \, {\ check {g}} (x) = \ int _ {\ widehat {G}} g ( \ chi) \ chi (x) \; d {\ widehat {\ lambda}} (\ chi)}$.

Is then such that the Fourier transform in , so obtained by means of this inversion formula from again back, at least up to a constant factor. This constant factor is due to the fact that the hair size is only unique up to one constant factor. Even in the prototypical case of real numbers, the well-known factor occurs when the Lebesgue measure is used on the group and the dual group. ${\ displaystyle f \ in L ^ {1} (G)}$${\ displaystyle {\ widehat {f}}}$${\ displaystyle L ^ {1} ({\ widehat {G}})}$${\ displaystyle {\ widehat {f}}}$${\ displaystyle f}$${\ displaystyle 2 \ pi}$

Fourier series

A feature on the circular group can in an obvious manner as a -periodic function to be construed, to put this . There , the Fourier transform is from a function to : ${\ displaystyle F}$${\ displaystyle \ mathbb {T}}$${\ displaystyle 2 \ pi}$${\ displaystyle f}$${\ displaystyle \ mathbb {R}}$${\ displaystyle f (x) = F (e ^ {ix})}$${\ displaystyle {\ widehat {\ mathbb {T}}} \ cong \ mathbb {Z}}$${\ displaystyle F}$${\ displaystyle \ mathbb {Z}}$

${\ displaystyle {\ widehat {F}} (n) = {\ frac {1} {2 \ pi}} \ int _ {\ mathbb {T}} F (z) z ^ {- n} d \ lambda ( z) = {\ frac {1} {2 \ pi}} \ int _ {0} ^ {2 \ pi} f (x) e ^ {- inx} dx}$

We see here the Fourier coefficients of . The Fourier inverse formula then leads to the known Fourier series . The abstract harmonic analysis thus provides the framework for a joint theoretical consideration of both the classical Fourier transformation and the Fourier series development. ${\ displaystyle f}$

Gelfand representation

Let G again be a locally compact Abelian group with Haar measure . The Fourier transform can also be interpreted in the following way. Each character defined by the formula ${\ displaystyle \ lambda}$${\ displaystyle \ chi \ in {\ widehat {G}}}$

${\ displaystyle \ phi _ {\ chi} (f): = \ int _ {G} f (x) {\ overline {\ chi (x)}} \; d \ lambda (x)}$

a continuous, linear multiplicative functional on . The Fourier transform turns out to be the Gelfand transform of the commutative Banach algebra . ${\ displaystyle \ phi _ {\ chi}}$${\ displaystyle L ^ {1} (G)}$ ${\ displaystyle L ^ {1} (G)}$

Non-Abelian groups

For non-Abelian groups it is no longer sufficient to look at the characters of the group; instead, one looks at unitary representations on Hilbert spaces . So be a locally compact topological group. A unitary representation of on a Hilbert space is now a continuous group homomorphism , where the unitary group denotes, equipped with the weak operator topology , which in this case corresponds to the strong operator topology . If there is now a Unterhilbert space of , so that still for all , then the representation can be restricted to, is called invariant subspace of the representation. A representation for which no non-trivial invariant subspace exists is called irreducible . One now chooses a representative system of the irreducible representations of a group with respect to unitary equivalence . In the Abelian case, this corresponds to the characters. Since any such representation can in a certain canonical way be extended to an algebra representation by adding ${\ displaystyle G}$ ${\ displaystyle \ pi}$${\ displaystyle G}$${\ displaystyle H _ {\ pi}}$ ${\ displaystyle \ pi \ colon G \ to U (H _ {\ pi})}$${\ displaystyle U (H _ {\ pi})}$ ${\ displaystyle V}$${\ displaystyle H _ {\ pi}}$${\ displaystyle g \ in G}$${\ displaystyle \ pi (g) (V) \ subseteq V}$${\ displaystyle U (V)}$${\ displaystyle V}$${\ displaystyle {\ hat {G}}}$${\ displaystyle \ pi}$${\ displaystyle L ^ {1} (G)}$

${\ displaystyle \ pi (f) = \ int _ {G} \ pi (x) f (x) \ mathrm {d} x}$

Set in a suitable sense of integration, can be used by the family ${\ displaystyle f \ in L ^ {1} (G)}$

${\ displaystyle {\ hat {f}} = (\ pi (f)) _ {\ pi \ in {\ hat {G}}}}$

define what is called the Fourier transform .

Further theorems of harmonic analysis deal with how and when and the space of the can be equipped with suitable structures that are obtained from the Fourier transformation (similar to the statement of the Plancherel formula ), whereby the Fourier transformation can be reversed. Such a result could not be obtained for all locally compact topological groups. ${\ displaystyle {\ hat {G}}}$${\ displaystyle {\ hat {f}}}$

Compact groups

A far-reaching generalization of the Fourier transform to compact groups is provided by Peter-Weyl's theorem. This theorem is particularly elementary, since the structure of is "discrete" in a certain sense (in the Abelian compact case actually discrete as topological space ) and can simply be understood as the orthogonal sum of matrices. ${\ displaystyle {\ hat {G}}}$${\ displaystyle {\ hat {f}}}$

Plancherel dimension for unimodular groups

In the event that the group is unimodular and two-countable and has a certain representation-theoretical property ( type 1 group , i.e. the group C * algebra is postliminal ), the Plancherel measure can be provided with regard to this measure Form a direct integral of the respective spaces of Hilbert-Schmidt operators , the Fourier transforms can then be understood and back-transformed as elements of this space . ${\ displaystyle {\ hat {G}}}$${\ displaystyle {\ hat {f}}}$

With regard to the Plancherel measure, sets of individual points can have a positive measure; these form the so-called discrete series , irreducible partial representations of the regular representation of the group. This is the case with compact groups, which in turn results in Peter-Weyl's theorem.

Non-unimodular groups

The inverse transformation is no longer possible in the same way on non-unimodular groups. In some cases , this can be remedied by special semi-invariant operators , that is, certain, generally only densely defined and unrestricted , positive , self-adjoint closed operators with which the are scaled in such a way that the Plancherel measure can be given, the Fourier transforms get a Hilbert space structure and an inverse transform becomes possible. These semi-invariant operators replace the ( equivariant ) constants that are necessary for scaling in the unimodular case, and are called Duflo-Moore operators or formal degree operators . ${\ displaystyle \ pi (f)}$${\ displaystyle {\ hat {G}}}$