On interpolation of interpolating sequences

Let A be a uniform algebra on the compact space X and σ a probability measure on X. We define the Hardy spaces HP(σ) and the HP(σ) interpolating sequences S in the p-spectrum Mp of σ. Under some structural hypotheses on (A, σ), we prove that if a sequence S ⊂ Mp is HP(σ) interpolating, then it is Hs(σ) interpolating for s < p. In the special case of the unit ball B of ℂn this answers a natural question asked in [8].