Fourier coefficients of continuous functions on compact groups

Abstract

Let G G be an infinite compact group with dual object Σ \Sigma . Letting K σ {\mathcal {K}_\sigma } be the representation space for σ ∈ Σ \sigma \in \Sigma , E 2 ( Σ ) {\mathcal {E}^2}(\Sigma ) is the set { A = ( A σ ) ∈ Π B ( K σ ) : ‖ A ‖ 2 2 = ∑ σ d σ Tr ⁡ ( A σ A σ ∗ ) > ∞ } \{ A = ({A^\sigma }) \in \Pi \mathcal {B}({\mathcal {K}_\sigma }):\left \| A \right \|_2^2 = {\sum _\sigma }{d_\sigma }\operatorname {Tr} ({A^\sigma }{A^{\sigma *}}) > \infty \} . For A ∈ E 2 ( Σ ) A \in {\mathcal {E}^2}(\Sigma ) , we show that there is a function f f in C ( G ) C(G) such that ‖ f ‖ ∞ ⩽ C ‖ A ‖ 2 {\left \| f \right \|_\infty } \leqslant C{\left \| A \right \|_2} and Tr ⁡ ( f ^ ( σ ) f ^ ( σ ) ∗ ) ⩾ Tr ⁡ ( A σ A σ ∗ ) \operatorname {Tr} (\hat f(\sigma )\hat f{(\sigma )^*}) \geqslant \operatorname {Tr} ({A^\sigma }{A^{\sigma *}}) for every σ ∈ Σ \sigma \in \Sigma .