CONNECTION BETWEEN THE LOWER P-FRAME CONDITION AND EXISTENCE OF RECONSTRUCTION FORMULAS IN A BANACH SPACE AND ITS DUAL

Abstract

In the present paper it is proved that under an additional assumption (which is automatically satisfied in case $p=2$) validity of the lower $p$-frame condition for a sequence $\{g_i\}\subset X^{\ast }$ implies that for $f$ in a subset of $X$ there exists a representation $f=\sum g_i(f)f_i$, where $\{f_i\}\subset X$ satisfies the upper $q$-frame condition, $\frac{1}{q}+\frac{1}{p}=1$. An example showing that the above representation is not necessarily valid for all $f$ in $X$ (neither reconstruction formula of type $g=\sum g(f_i)g_i$ for all $g\in X^{\ast }$) is given. It is shown that when $\mathcal{D}(U)$ is dense in $X$, $g\in X^{\ast }$ can be represented as $g=\sum g(f_i)g_i$ if and only if $\sum g(f_i)g_i$ converges.