Almost Unimodular Systems on Compact Groups with Disjoint Spectra

We emulate the Rademacher functions on any non-commutative compact group requiring the resulting system to have pairwise disjoint spectra. Introduction Let G be a compact topological group with normalized Haar measure μ and dual object Ĝ. Let π : G → U(Hπ) be an irreducible unitary representation of G. Then, given an integrable function f : G → C, the Fourier coefficient of f at π is defined as follows f̂(π) = ∫ G f(g)π(g) dμ(g) ∈ B(Hπ). It is well-known how to construct n functions f1, f2, . . . fn : G → C satisfying (P1) f1, f2, . . . , fn have pairwise disjoint supports on G. (P2) The norm of f̂k(π) on Sπ q does not depend on k for any π ∈ Ĝ. Just take disjoint translations of a common function on G with sufficiently small support. Also, one can consider the dual properties with respect to the Fourier transform on G. Namely, (P̂1) f̂1, f̂2, . . . , f̂n have pairwise disjoint supports on Ĝ. (P̂2) The absolute value |fk(g)| does not depend on k for almost all g ∈ G. If G is abelian we can easily construct such a system by taking n irreducible characters of G. Moreover, many other constructions are available. Namely, we can take n disjoint translations of a common function on Ĝ. Then we get the desired property by Pontrjagin duality. This kind of systems have been applied to study the Fourier type constants of finite-dimensional Lebesgue spaces: if 1 ≤ p < q ≤ 2, the Fourier q-type constant of lp(n) with respect to Ĝ is n , which is optimal among the family of n dimensional Banach spaces. In particular, this provides sharp results about the Fourier type of infinite-dimensional Lebesgue spaces. However, irreducible characters on non-commutative compact groups are no longer unimodular. Furthermore, the dual object is not a group anymore and consequently Pontrjagin duality does not hold. Besides, Tannaka’s theorem (the non-commutative counterpart of Pontrjagin theorem) does not fit properly in this context. Hence, it is natural to wonder if it is possible to construct a system Φ = { fm : G → C ∣∣ m ≥ 1 } made up of functions f1, f2, . . . satisfying similar properties to (P̂1) and (P̂2). In this paper, we construct an almost unimodular system of trigonometric polynomials 2000 Mathematics Subject Classification: 43A77. Partially supported by the Project BFM 2001/0189, Spain. 1