arXiv subjects

M. H. Safaryan

Publications and source records attributed to M. H. Safaryan.

2 recordsLinked to original sources

arXiv2013

On generalizations of Fatou's theorem for the integrals with general kernels

We define $λ(r)$-convergence, which is a generalization of nontangential convergence in the unit disc. We prove Fatou-type theorems on almost everywhere nontangential convergence of Poisson-Stiltjes integrals for general kernels $\{φ_r\}$, forming an approximation of identity. We prove that the bound \md0 \limsup_{r\to 1}λ(r) \|φ_r\|_\infty<\infty \emd is necessary and sufficient for almost everywhere $λ(r)$-convergence of the integrals \md0 \int_\ZT φ_r(t-x)dμ(t). \emd

math.CA↗

arXiv2013

On a theorem of Littlewood

In 1927 Littlewood constructed an example of bounded holomorphic function on the unit disk, which diverges almost everywhere along rotated copies of any given curve in the unit disk ending tangentially to the boundary. This theorem was the complement of a positive theorem of Fatou 1906, establishing almost everywhere nontangential convergence of bounded holomorphic functions. There are several generalizations of the Littlewood's theorem which proofs are based on the specific properties of Poisson kernel. We generalize Littlewood's theorem for operators having general kernels.

math.CA↗