We show that if f(z) = ∑ anz n is a holomorphic function in the Dirichlet space of the unit disk, then almost all of its randomizations ∑ ±anz are multipliers of that space. This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip1/2 of functions obeying a Lipschitz condition with ex...

It is well-known that Littlewood–Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p (\mathbb {R})$ for all $1 \lt p \infty $. In this note it shown $$ \| S_{\mathcal {I}_{E_2}} \|_{L^p {R}) \right

A subset C of an abelian group G is said to be a minimal additive complement W⊆G if C+W=G and C′+W≠G for any proper C′⊂C. In this paper, we study certain subsets the integers that arise or do not as complements. Following suggestion Kwon, show bounded-below sets with arbitrarily large gaps Moreover, our construction shows such set belongs co-minimal pair, strengthening result Biswas Saha lacuna...

Marouf (1993) presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. Patterson (2003), extended those concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for non-negative sum ability matrices. In Patterson and Savas (2006) extended the definitions presented in (Patterson, 2003) to l...