For various $L^p$-spaces (1 ≤ p < ∞) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For μ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in $L^p\left(\mu \right)$. For M a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function that generates...

Linear isometries of N p(D) onto N p(D) are described, where N p(D), p > 1, is the set of all holomorphic functions f on the upper half plane D = {z ∈ ℂ: Im z > 0} such that supy>0 ∫ℝ lnp (1 + |(x + iy)|) dx < +∞. Our result is an improvement of the results by D.A. Efimov.

We prove precise estimates for the diametral dimension of certain weighted spaces of germs of holomorphic functions defined on strips near ℝ. This implies a full isomorphic classification for these spaces including the Gelfand-Shilov spaces $S{¹}_{\alpha }$ and $S{₁}^{\alpha }$ for α > 0. Moreover we show that the classical spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions are not isomorphic.

In this paper we consider several conditions for sequences of points in M(H ∞) and establish relations between them. We show that every interpolating sequence for QA of nontrivial points in the corona $$M\left(H^{\infty }\right)\setminus 𝔻$$ of H ∞ is a thin sequence for H ∞, which satisfies an additional topological condition. The discrete sequences in the Shilov boundary of H ∞ necessarily satisfy the same condition.