The space Hp, 1 ≤ p ≤ ∞ consists of those analytic functions f(z) regular in the unit circle, for which Mp (f;r) is bounded for O ≤ r ≤ 1, where

These spaces have been extensively studied.

One well known result concerning these spaces is that if f(z) = Σ ∞n=0 anzn and {an} ɛ lp for some p, 1 ≤ p ≤ 2, then f ɛ Hq, where p-1+q-1 = 1, and conversely if f ɛ Hp, 1 ≤ p ≤ 2, then {an} ɛ lq. We propose to generalize this result to deal with functions f(z) = Σ ∞n=0 anzn with {n-λ an; n = 1, 2,...} ɛ lp, where λ ≥ 0. The resulting generalization is contained in the theorems below.

However, in order to make these generalizations we must first generalize the spaces Hp. To this end we make the following definition.