$\mathcal {N}_p$-Type Functions with Hadamard Gaps in the Unit Ball $\mathbb {B}$

Abstract

A function $f\in \mbox{Hol}(\mathbb {B})$ written in the form

$$\displaystyle f(z)=\sum _{k=0}^\infty P_{n_k}(z), $$

where $P_{n_k}$ is a homogeneous polynomial of degree $n_k$, is said to have Hadamard gaps if, for some $c>1$ ,

$$\displaystyle \frac {n_{k+1}}{n_k}\ge c $$

for all $k \geq 0$. Hadamard gap series on spaces of holomorphic functions in $\mathbb {D}$ or in $\mathbb {B}$ has been extensively studied. In this chapter, we study some aspects of Hadamard gap series in $\mathcal {N}_p$-spaces.