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

ЗаглавиеCONNECTION BETWEEN THE LOWER P-FRAME CONDITION AND EXISTENCE OF RECONSTRUCTION FORMULAS IN A BANACH SPACE AND ITS DUAL
Вид публикацияJournal Article
Година на публикуване2005
АвториStoeva D
СписаниеAnnuaire de l’Université de Sofia “St. Kliment Ohridski”. Faculté de Mathématiques et Informatique
Том97
ключови думиBanach spaces, dual spaces, lower bound, p-frames, reconstructions
Резюме

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 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 $\mathcal{D}(U)$ 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.

2000 MSC

42C15, 40A05

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