On the Non-Equivalence of Rearranged Walsh and Trigonometric Systems in Lp

Both the Walsh system and the trigonometric system are systems of characters on a compact abelian group. This explains that many of the results in the theory of those systems are parallel. However, those similarities do usually not extend to the case when the systems are compared directly. So it is known that the Walsh system in the Walsh-Paley order and the trigonometric system are not equivalent in Lp for p 6= 2, see [5]. A “power-type” non-equivalence for those systems was recently shown in [4]. It does not seem natural to fix the order of the systems in this basis equivalence problem. In [4] the conjecture was made that non-equivalence also holds for arbitrary rearrangements of the Walsh system. Nevertheless, the methods used in that paper are very particular to the case of the Walsh-Paley order. The aim of this note is to address the more general equivalence problem. In a first part, we relate the equivalence question for a fixed ordering to a question of algebraic combinatorial type. In a second part, we apply this approach to prove non-equivalence for a number of orderings. We obtain estimates of power type but we do not attempt to find the optimal estimates here.