Hardy space and Bergman space on the Octonions

In this paper, square-integrable Octonion-valued function spaces on R³ and R⁷ are decomposed into the direct sum of Octonion Hardy and conjugate Hardy spaces, and square-integrable Octonion function spaces on the upper half spaces R ₊ ⁴ and R ₊ ⁸ are decomposed into infinity direct sum of subspaces in which the first components are just the Octonion Bergman spaces. The Octonion Hardy Spaces on R³ and R⁷ are characterized.