# Atomic decomposition from S-functions

• Akihito Uchiyama
Part of the Springer Monographs in Mathematics book series (SMM)

## 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, fSand
$$\sum\limits_{i = 1}^J {\left\| {S_{\varphi i,\delta } f} \right\|_{L^p } } < + \infty .$$
(5.3)
Then, there exists a polynomial P(x) such that
$$\left\| {f - P} \right\|_{H^p } \le C\left( {\left\{ {\varphi _1 , \cdots ,\varphi _J } \right\},\delta ,p} \right)\sum\limits_{i = 1}^J {\left\| {S_{\varphi i,\delta } f} \right\|_{L^p } .}$$

## Keywords

Euclidean Space Number Theory Algebraic Geometry Hardy Space Important Paper
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.