Connection between the lower P-frame condition and existence of reconstruction formulas in a Banach space and its dual

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 $\{g_i\}\subset X^*$ 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 \\[3pt] 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 $g \in X^*$) is given. It is shown that when $\du$ is dense in $X$, $g\in X^*$ can be represented as $g=\sum g(f_i) g_i$ if and only if $\sum g(f_i) g_i$ converges.

Annotation in PDF format: (PDF document 147Kb)