On Banach frameness of degenerate weighted exponential system

This work deals with the frameness of weighted exponential system $E(\omega ,Z)= \left \{\omega (t)e^{int} \right \}_{n\in Z} $ in the space $L_{p} (-\pi ,\pi )$ , $p>1$ , with the weight function $\omega (t)$ of general form. Basis properties of $E(\omega ,Z)$ in $L_{p} (-\pi ,\pi )$ , $p>1$ , are studied, in other words, the criteria of completeness, minimality and basicity of the system $E(\omega ,Z)$ in the space $L_{p} (-\pi ,\pi )$ , $p>1$ , are given. Sufficient conditions for the completeness and minimality of $E(\omega ,Z\backslash F)$ in $L_{p} (-\pi ,\pi )$ , $p>1$ , are found, where F is an arbitrary finite nonempty subset of the set of integers Z. A different method to prove that the system $E(\omega ,Z\backslash F)$ does not form a Schauder basis for $L_{p} (-\pi ,\pi )$ , $p>1$ , is given. Theorem on a property of expansion system and criterion of Banach frameness for $E(\omega ,Z)$ in $L_{p} (-\pi ,\pi )$ , $p>1$ , are proved. In particular, it is proved that the system $E(\omega ,Z)$ with defect cannot form atomic decomposition for $L_{p} (-\pi ,\pi )$ , $p>1$ . The obtained results are the generalizations of those on the atomic decomposition of power weighted exponential system in $L_{p} (-\pi ,\pi )$ , $p>1$ , and the frameness of weighted exponential system in $L_{2} (-\pi ,\pi )$ .