Atomic decomposition from S-functions

Abstract

In this section we prove the following: Theorem 5.1. Let

$$ \left\{ {\varphi _1 ,\varphi _2 , \cdots ,\varphi _J } \right\} \subset S $$

(5.1)

and

$$ \sup \left\{ {\sum\limits_{i = 1}^J {\left| {F\varphi _i \left( {t\xi } \right)} \right|} :t > 0} \right\} > 0{\rm{ }}for{\rm{ }}any{\rm{ }}\xi \in {\rm{R}}^n \backslash \left\{ 0 \right\}. $$

(5.2)

Let p ∈ (0,1], δ>0, f ∈ S′ and

$$ \sum\limits_{i = 1}^J {\left\| {S_{\varphi i,\delta } f} \right\|_{L^p } } < + \infty . $$

(5.3)

Then, there exists a polynomial P(x) such that

$$ \left\| {f - P} \right\|_{H^p } \le C\left( {\left\{ {\varphi _1 , \cdots ,\varphi _J } \right\},\delta ,p} \right)\sum\limits_{i = 1}^J {\left\| {S_{\varphi i,\delta } f} \right\|_{L^p } .} $$