Nonnegative functions in weighted hardy spaces
Preprint Series # 653
Inoue, Jyunji and Nakazi, Takahiko Nonnegative functions in weighted hardy spaces. (2004);
Let $W$ be a nonnegative summable function whose logarithm is also summable with respect to the Lebesgue measure on the unit circle. For $0 < p < \infty,~H^p(W)$ denotes a weighted Hardy space on the unit circle. When $W \equiv 1,~H^p(W)$ is the usual Hardy space $H^p$. We are interested in $H^p(W)_+$ the set of all nonnegative functions in $H^p(W)$. If $p \geq 1/2,~H^p_+$ consists of constant functions. However $H^p(W)_+$ contains a nonconstant nonnegative function for some weight $W$. In this paper, if $p \geq 1/2$ we determine $W$ and describe $H^p(W)_+$ when the linear span of $H^p(W)_+$ is of finite dimension. Moreover we show that the linear span of $H^p(W)_+$ is of infinite dimension for arbitrary weight $W$ when $0 < p < 1/2$.