Characterization of Abelian groups with spectral analysis and spectral synthesis

In [12] M. Laczkovich and G. Székelyhidi proved the following result.

Theorem 5.1. Spectral analysis holds on an Abelian group if and only if its torsion free rank is less than the continuum.

150 L. Székelyhidi According to Theorem 3.4 there are Abelian groups on which spectral analysis holds and spectral synthesis fails to hold: for instance, any Abelian group with torsion free rank ω, like Z^ω above. On the other hand, a complete description of those Abelian groups on which spectral synthesis holds is still missing. The conjecture formulated in the previous section has neither been proved nor disproved yet. An interesting situation can be presented by the additive group of rational numbers. It is not known if spectral synthesis holds on this group. Actually, this group is not finitely generated, however, its torsion free rank is 1. If spectral synthesis does not hold on the rationals, then the above conjecture is drastically disproved: for Abelian groups with torsion free rank zero spectral synthesis holds, as these are exactly the torsion groups. The next simplest case is obviously the case of torsion free rank 1. On the other hand, if spectral synthesis holds on the rational group, then this is the first example for a torsion free group where spectral synthesis holds and the group is not finitely generated.

In addition to the above conjecture in [13] we proved the following theorem.

Theorem 5.2. The torsion free rank of any Abelian group is equal to the dimen-sion of the linear space consisting of all complex additive functions of the group in the sense that either both are finite and equal, or both are infinite.

Proof. LetG be an Abelian group and let let k=r0(G)6+∞. ThenG has a subgroup isomorphic toZ^k. Ifk is infinite then this is equal to the non-complete direct product ofkcopies ofZ. We will identify this subgroup withZ^k. Obviously Z^k has at least k linearly independent complex additive functions; for instance we can take the projections onto the different factors of the product group. On the other hand, by the above mentioned result in [3] any homomorphism of a subgroup of an Abelian group into a divisible Abelian group can be extended to a homomorphism of the whole group. As the additive group of complex numbers is obviously divisible, the above mentioned linearly independent complex additive functions ofZ^kcan be extended to complex homomorphisms of the whole groupG, and the extensions are clearly linearly independent, too. Hence the dimension of the linear space of all complex additive functions ofGis not less then the torsion free rank ofG.

Now we suppose thatk <+∞. LetΦdenote the natural homomorphism ofG onto the factor group with respect toZ^k. As it is a torsion group, hence for each elementg ofGthere is a positive integer nsuch that

0 =nΦ(g) = Φ(ng),

thusngbelongs to the kernel ofΦ, which isZ^k. This means that there exist integers m₁, m₂, . . . , m_k such that

ng= (m1, m2, . . . , mk).

Suppose now that there arek+ 1linearly independent complex additive functions a1, a2, . . . , ak+1onG. Then there exist elementsg1, g2, . . . , gk+1inGsuch that the

Discrete spectral synthesis 151 (k+ 1)×(k+ 1)matrix¡

ai(gj)¢

is regular. Forl= 1,2, . . . , kwe let el denote the vector inC^kwhosel-th coordinate is1, the others are0. By our above consideration there are integersm^(j)_l ,nj forl= 1,2, . . . , kandj= 1,2, . . . , k+ 1 such that

n_jg_j= (m^(j)₁ , m^(j)₂ , . . . , m^(j)_k ). Hence we have

ai(njgj) =ai(m^(j)₁ , m^(j)₂ , . . . , m^(j)_k ) =

=m^(j)₁ ai(e1) +m^(j)₂ ai(e2) +· · ·+m^(j)_k ai(ek), and therefore

ai(gj) = Xk

l=1

m^(j)_l n_j ai(el)

holds fori, j= 1,2, . . . , k+ 1. This means that the linearly independent columns of the matrix¡

ai(gj)¢

are linear combinations of the columns of the matrix ¡ ai(el)¢ for i = 1,2, . . . , k+ 1; l = 1,2, . . . , k. But this is impossible, because the latter matrix has onlykcolumns, hence its rank is at mostk.

We have shown that if the torsion free rank of G is the finite numberk then the dimension of the linear space consisting of all complex additive functions ofG

is at mostk, hence the theorem is proved. ¤

Another characterization of Abelian groups with finite torsion free rank is given by the following result (see [13]).

Theorem 5.3. The torsion free rank of an Abelian group is finite if and only if any complex bi-additive function is a bilinear function of complex additive functions.

Hence our conjecture has two more equivalent formulations:

- Spectral synthesis holds on an Abelian group if and only if there are only finitely many linearly independent additive functions on the group.

- Spectral synthesis holds on an Abelian group if and only if any complex bi-additive function is a bilinear function of complex bi-additive functions.

References

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[3] Hewitt, E., Ross, K.,Abstract Harmonic Analysis I.,II., Springer Verlag, Berlin, 1963.

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Cambridge Phil. Soc., Vol. 61 (1965), 617–620.

152 L. Székelyhidi [5] Djokovič, D. Z., A representation theorem for(X1−1)(X2−1). . .(Xn−1)and

its applications,Ann. Polon. Math., Vol. 22 (1969), 189–198.

[6] Székelyhidi, L., Convolution type functional equations on topological Abelian groups, World Scientific Publishing Co. Pte. Ltd., Singapore, New Jersey, London, Hong Kong, 1991.

[7] Székelyhidi, L., On the extension of exponential polynomials, Math. Bohemica, vol. 125, no. 3, pp. 365–370, 2000.

[8] Székelyhidi, G., Spectral Synthesis on Locally Compact Abelian Groups, (essay), Cambridge, Trinity College, 2001.

[9] Székelyhidi, L., A Wiener Tauberian theorem on discrete abelian torsion groups, Annales Acad. Paedag. Cracov., Studia Mathematica I., Vol. 4 (2001), 147–150.

[10] Székelyhidi, L.,The failure of spectral synthesis on some types of discrete Abelian groups,Jour. Math. Anal. Appl., Vol. 291 (2004), 757–763.

[11] Bereczky, A., Székelyhidi, L.,Spectral synthesis on torsion groups,Jour. Math.

Anal. Appl., Vol. 304/2, (2005) 607–613.

[12] Laczkovich, M., Székelyhidi, G.,Harmonic analysis on discrete Abelian groups, to appear inProc. Amer. Math. Soc.

[13] Székelyhidi, L.,Polynomial functions and spectral synthesis, to appear in Aequa-tiones Math.

László Székelyhidi Institute of Mathematics University of Debrecen

4010 Debrecen, Pf. 12. Hungary

Annales Mathematicae et Informaticae 32(2005) pp. 153–157.

Note on symmetric alteration of knots of