Analyzes harmonic on a finished abelian group

In Mathematical, the analyzes harmonic on an abelian group finished is a particular case of analyzes harmonic corresponding if the group is abelian and finished.

The harmonic analysis makes it possible to define the concept of Transformée of Fourier or the Produit convolution. It is the framework many theorems like that of Plancherel, the equality of Parseval or the Dualité of Pontryagin.

The case where the group abelian and is finished is simplest of the theory, the transform of Fourier is limited to a finished sum and the dual group is isomorphous with the group of origin.

The harmonic analysis on a finished abelian group has many applications, particularly in modular Arithmétique and Information theory.

Context

Algebra of the group

See also: Algebra of a group finished

The harmonic analysis constitutes a tool of study of the space of the applications C ^G of a unit, here an abelian group finished G (noted in all the article additivement), in the body of the complex numbers C . This space has several structures. Initially, as C is a body, C ^G is a vector Space complex of Dimension G if G indicates the order of the group G . It is naturally provided with a square product < | > defined by:

\ forall F, H \ in \ mathbb C^G = \ frac 1g \ sum_ {S \ in G} F (S) ^*. \, H (S) \;

Here, and in the remainder of the article if Z indicates a complex number, Z ^* indicates its Conjugué. This square product said canonical , confers on C ^G a structure of Espace of Hilbert, noted L ² ( G ).

In all the article ( E _s) where S described G , indicates the canonical base of C ^G, i.e. E _s indicates the function which with T element of G associates 0 except if T is equal to S and then E _s ( S ) = 1.

The vector space generated by the family ( E _s) is provided with the following internal multiplication, prolonging that of the group G :

\ forall (a_s) _ {S \ in G} \; (b_t) _ {T \ in G} \ in \ mathbb C^G \ quad (\ sum_ {S \ in G} a_s.e_s) (\ sum_ {T \ in G} b_t.e_t) = \ sum_ {S, T \ in G} a_sb_t.e_ {St} \;

This multiplication confers on L ² ( G ) a structure of semi-simple Algèbre, in general noted C .

The theory of the harmonic analysis on a finished abelian group uses indifferently the notations L ² ( G ) or C to indicate the basic structure of the theory. In this article the notations used are those of C . Thus, if has is an element of the algebra, one uses the notation here has _s to indicate the coordinate of has in the canonical base, this notation corresponds to the equality has _s = has ( S ) if has is regarded as an element of L ² ( G ).

Dual group

See also: Character of a group finished

The dual group of G , noted here $\ scriptstyle \ widehat G$ is consisted of the whole of S characters of G . It forms an isomorphous group with G . It consists of applications of C in G , therefore is included in L ² ( G ) identified here with C . It forms in fact a orthonormal Base of the algebra.

The algebra of the dual group is canonically isomorphous with the whole of the applications of the dual group in C . These applications are prolonged by linearity in an application which with a linear combination of character associates a complex, i.e. with an element of dual of the algebra C . The dual one of C is thus canonically isomorphous with the algebra of the dual group of G .

Theory of the harmonic analysis

Transform of Fourier

See also: Transform of Fourier

The equality of Parseval in the case of a space of finished size watch which any element has C checks the following equality:

a = \ sum_ {S \ in G} a_se_s= \ sum_ {\ chi \ in \ widehat G} a_ {\ chi} \ chi \ quad with \ quad a_ {\ chi} = = \ frac 1g \ sum_ {S \ in G} a_s^*. \ chi (S) \ quad and \ quad = \ sum_ {\ chi \ in \ widehat G} |a_{\chi}|^2 \;

Here (a_s) indicates the coordinates of has in the canonical base and (a_χ) the coordinates of has in the base of the characters.

* the Transformée of Fourier of an element has C corresponds to the generally noted function $\ scriptstyle \ widehat has$ dual group of G in C , i.e. a function which with a character of the group associates a complex, definite by:

\ widehat has (\ chi) = \ frac {1} {\ sqrt G} \ sum_ {S \ in G} a_s \ chi (S) ^* \;

* the transform of Fourier is a linear application of the algebra of G in its dual.

Equality of Parseval

See also: Equality of Parseval

The square product generates a canonical isometry between the algebra of G and its dual. It is thus possible to identify them, in this context, the following property is checked:

* the transform of Fourier on the group G is a linear Isométrie of the algebra of the group G in the algebra of its dual what results in the following equality said of Parseval:

\ forall has, B \ in \ mathbb C \ quad_ {\ mathbb C} =< \ hat has|\ hat b>_ {\ mathbb C G}

Formulate of Plancherel

See also: Theorem of Plancherel

* the formula following, known as of inversion of Plancherel, is checked.

\ forall has \ in \ mathbb C \ quad has = \ frac 1 {\ sqrt G} \ sum_ {\ chi \ in \ widehat G} \ widehat has (\ chi) \ chi \;

Indeed, the square products of each of the two members of the equality by the same character are equal:

\ forall \ zeta \ in \ widehat G \ quad < \ frac 1 {\ sqrt G} \ sum_ {\ chi \ in \ widehat G} \ widehat has (\ chi) \ chi|\ zeta> = \ frac 1 {\ sqrt G} \ widehat has (\ zeta) ^* = \ frac 1g \ sum_ {S \ in G} a_s^* \ zeta (S) =

Product of convolution

See also: Produces convolution

The product of convolution is defined simply in this context:

* Is has and B two elements of the algebra of the group G having for coordinates ( has _s) and ( B _s), the produces convolution of has and of B , noted * B has, is the element of the algebra having the coordinates (c_s) defined by:

c_s= \ sum_ {T \ in G} a_tb_ {St} \ quad thus \ quad a*b = \ sum_ {S \ in G} c_s. E. _s= \ sum_ {S, T \ in G} a_tb_ {St} e_s
\;

One has the following proposal:

* Is has and B two elements of the algebra of the group G , the transform of Fourier of has * B is the product of the transforms of Fourier of has and of B .

\ forall has, B \ in \ mathbb C \ quad \ widehat {a*b} \, (\ chi) = \ hat has (\ chi). \ hat B (\ chi) \;

Indeed, if χ is a character of the group:

\ widehat {a*b} (\ chi) = \ frac 1g \ sum_ {S, T \ in G} a_tb_ {St} \ chi (S) ^* \;

If one notes U the value S - T , one obtains:

\ widehat {a*b} (\ chi) = \ frac 1g \ sum_ {T, U \ in G} a_tb_u \ chi (t+u) ^*=
\ Big (\ frac 1 {\ sqrt G} \ sum_ {T \ in G} a_t \ chi (T) ^* \ Big). \ Big (\ frac 1 {\ sqrt G} \ sum_ {U \ in G} b_u \ chi (U) ^* \ Big) = 
\ hat has (\ chi). \ hat B (\ chi) \;

One from of deduced the usual properties from the product of convolution:

* the product of convolution is an internal operation of the algebra of the group commutative, associative, distributive compared to the addition.

One can express these properties in the following way:

* the structure ( C , +, *) is a semi-simple Algèbre isomorphous with the algebra of dual of G and thus with C .

Indeed, it is enough to notice that G and its dual is isomorphous.

Duality of Pontryagin

See also: Duality of Pontryagin

* Is H a sub-group of G , one calls orthogonal group H , often noted

\ scriptstyle H^ {\ perp}

, the sub-group of the dual group of G definite in the following way:

H^ {\ perp} = \ {\ chi \ in \ widehat G \ quad/\ quad \ forall H \ in H \ quad \ chi (H) = 1 \}

The duality of Pontryagin is expressed through the three following properties:

* G and its bidual is canonically isomorphous.

* the dual one of the quotient G / H is isomorphous with orthogonal of H .

* the dual one of H is isomorphous with the quotient of dual of G by the orthogonal one of H .

Formulate sommatoire of Poisson

See also: Formula sommatoire of Poisson

In this paragraph H designates a sub-group of G , H its order and K the order of the orthogonal group of H . The equality H . K = G is thus checked. One notes has an element of the algebra of G and has _s its coordinates in the canonical base.

* the equality following, known as formula sommatoire of Poisson is checked:

$\ frac 1 {\ sqrt H} \ sum_ {T \ in H} a_t = \ frac 1 {\ sqrt K} \ sum_ {\ chi \ in H^ {\ perp}} \ hat has (\ chi) \;$

Applications

Arithmetic modular

See also: Arithmetic modular

The first historical uses of the characters aim at the arithmetic one. The Symbole of Legendre is an example of character on the multiplicative group of the Corps finished Z /p Z where Z indicates the ring of the relative whole and p a Prime number odd.

It is used for the calculation of the sums of Gauss or the periods of Gauss. This character is at the base of a demonstration of the quadratic Loi of reciprocity.

Symbol of Legendre

See also: Symbol of Legendre

In this paragraph p a prime number odd indicates (i.e. different from two). G is here the group '' Z '' /p '' Z ''. The symbol of Legendre indicates the function, which with an entirety has , associates 0 if has is a multiple of p , 1 if the class of has is a square different from 0 in Z /p Z and -1 if not.

* the image of the function symbol of Legendre on the multiplicative group of Z /p Z corresponds to the character with value as a whole {- 1, 1}.

Indeed, the symbol of Legendre is defined on Z . This function is constant on the classes of entireties modulo p , it is thus defined on the multiplicative group of '' Z '' /p '' Z ''. On this group, the symbol of Legendre takes its values as a whole {- 1, 1} and is a morphism of group, because the symbol of Legendre is a Caractère of Dirichlet.

the demonstrations are given in the associated article.

Summon Gauss

See also: Somme of Gauss

In the remainder of the article, F _p indicates the Corps finished of cardinal p or p is a prime number odd.

* Is ψ a character additive group ( F _p, +) and χ a character of the multiplicative group ( F _p^*.), then the nap of Gauss associated with χ and ψ are the complex number, here noted G (χ, ψ) and defined by:

G (\ chi, \ psi) = \ sum_ {X \ in F_p^*} \ chi (X). \ psi (X) \;

In term of Transformed of Fourier, one can consider the application which with χ associates G (χ, ψ^*) like the transform of Fourier of the prolongation of χ with F _p by the equality χ (0) = 0 in the additive group of the body and the application which with ψ associates G (χ^*, ψ) like transformed of Fourier of the restriction of ψ on F _p^* in the multiplicative group of the body.

The sums of Gauss are largely used into arithmetic, for example for the calculation of the periods of Gauss, they for example, to determine the sum of the values of the group of the quadratic residues of the roots p - ièmes of the unit and more generally to determine the roots of the cyclotomic Polynôme of index p .

Quadratic law of reciprocity

See also: quadratic Law of reciprocity

The sums of Gauss have an important historical application, the quadratic law of reciprocity, it is expressed in the following way:

* Is p and Q two prime numbers odd distinct, the following equality is checked:

\ left (\ frac {p} {Q} \ right) \ left (\ frac {Q} {p} \ right) = (- 1) ^ {\ frac {(p-1) (q-1)}{4}}

This theorem is shown in the article Somme of Gauss.

Character of Dirichlet

See also: Character of Dirichlet

To dismount the Theorem of the arithmetic progression, affirming that any invertible class of the Anneau Z/nZ contains an infinity of prime numbers, Dirichlet generalizes work of Gauss and systematically studies the group of the characters of the group of the unit of a quotient of Z .

The use of the transform of Fourier is a key stage of the demonstration. The characters of Dirichlet have a big role in the analytical Théorie of the numbers particularly to analyze the roots of the function ζ of Rieman.

Finished vector space

See also: harmonic Analysis on a vector space finished

A particular case is that of the vector spaces on a Corps finished. The properties of the finished bodies make it possible to establish the results of the theory in a slightly different form. This case is used for example in Information theory through the study of the Boolean functions, corresponding if the body contains two elements. The theory is used to solve questions of Cryptologie in particular for the boxes-S, like for the codings by flood. The harmonic analysis on a finished vector space also intervenes in the context of the Théorie of the codes and particularly for the linear codes, for example to establish the Identité of Mac Williams.

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