# Space of analytic function and sequence space $l^p$

Let $\mathbb{D} = \{z:|z|<1\}$ be open unit disc in complex plane. Define space of analytic function: $N^p=\{f:\mathbb{D} \to \mathbb{C} | f(z)=\sum_{n=0}^{\infty} a_n z^n, \sum_{n=0}^{\infty}|a_n|^p < \infty\}$. For $p=1$ we have Wiener algebra, and for $p=2$ we have Hardy space $H^2$. What we know about this space of analytic function for $p \in (1,2) \cup (2,\infty]$. Describe relationship between Hardy spaces and $N^p$ spaces? Is there any paper or reference for this kind of spaces?

• I think that rather than ask "where do I find a book about these spaces" one should ask "what do I want to know about these spaces"? That is: what kinds of questions do you have about how the summability of the Taylor series of an analytic function affects properties of the function? Mar 30, 2017 at 16:09
• What is specific name for these spaces? Is there any paper, reference... Mar 31, 2017 at 10:20
• OK, if you want to know how $N^p$ is related or unrelated to the scale of Hardy spaces, then you might find some things in Duren's book Theory of $H^p$-spaces. However, without further details of what kinds of question you want to answer or investigate, I'm not sure if that book will turn out to be helpful Mar 31, 2017 at 16:16