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The basic tools for the investigation of different algebraic and analytical structures are representation and duality. “Representation” means that we establish a correspondence between our abstract structure and a similar, more particular one. Usually this more particular structure, the “representing” structure is formed by functions, defined on a set which is the so-called “dual” object. In order to get a “faithful” representation, it seems to be reasonable that the correspondence in question is one-to-one. Another reasonable requirement is that if the same procedure is applied to the dual object, then its dual can be identified with the original structure. In order to do that, the dual object should have an “internal” characterization. Finally, a characterization of the “representing” structure is also desirable : which functions on the dual object belong to the “representing” structure?

In the case of the ideals of () we have seen that any proper ideal is included in a maximal ideal. We can prove the same for any proper regular ideal in any commutative algebra.

Let be an Abelian group. We say that is a if every element of has finite order. In other words, for every in there exists a positive integer with = 0. Hence is not a torsion group if and only if there exists an element of which generates a subgroup isomorphic to ℤ.

We recall from the previous part that for a given locally compact Abelian group any proper closed translation invariant subspace of () is called a variety. The set of all exponentials in a variety is called the of the variety, and the set of all exponential monomials in a variety is called the of the variety. If is a variety, then denotes the spectrum of and we write for τ (). If μ is in 蒙(), then we use the notation sp for the spectrum of the annihilator of the ideal generated by , and for any subset of 蒙() the spectrum, or spectral set of , is the spectrum, or the spectral set of the annihilator of the ideal generated by .

In Section 4.1 we have seen that the study of varieties in () is equivalent to the study of solution spaces of convolution type functional equations. As the solutions of convolution type functional equations are mean periodic functions, it seems to be reasonable to set them into the center of our investigations. We recall (see also [10]) that for a locally compact Abelian group the continuous function : → ℂ is called mean periodic if there exists a nonzero compactly supported complex Radon measure on such that

The result of Lefranc on spectral synthesis in (ℤ) can be used to give a simple method for the solution of linear systems of homogeneous difference equations with constant coefficients. The method is based on the simple fact that varieties in (ℤ) are exactly the solution spaces of such systems of equations. In the case = 1 the situation reduces to the classical theory of linear homogeneous difference equations with constant coefficients, as it has been exhibited in Section 2.4. Now we present a more detailed analysis of this subject in several variables (see [74]). First we recall and adjust our previous notation to the present situation.

In this section we formulate the basic problems of spectral analysis and spectral synthesis on commutative hypergroups and solve these problems on some types of hypergroups (see [72]). For more about -spectral synthesis on hypergroups we refer to the the paper [78]. In [7] a Wiener Tauberian Theorem is presented for commutative locally compact hypergroups, whose dual is a hypergroup under pointwise operations. For further references on -spectral synthesis in hypergroups the reader is referred to [8], [31], [50].

Let be a countable set equipped with the discrete topology and let be a positive integer. We consider a set () of polynomials in complex variables. If for any nonnegative integer the symbol denotes the set of all elements in for which the degree of is not greater than , then we suppose that the polynomials with in form a basis for all polynomials of degree not greater than .