On exact frames in topological algebras (Q2823069)

Let \(A\) denotes a real or complex commutative locally convex separable topological algebra. A countable sequence \(\{x_n \} \subset A\) is a frame if there exists a sequence \(\{f_n\}\) of linear functionals such that for each \(x \in A\) we have \(x = \lim_{n\rightarrow\infty} \sum_{i=1}^n f_i (x)x_i\). A frame \(\{x_n \}\) is said to be exact, if for all \(j \in \mathbb{N}\) the sequence \(\{x_n \}_{n\neq j}\) is not a frame. The paper discussed several results of the following type:NEWLINENEWLINEA frame \(\{x_n\}\) is exact, provided for each \(j \in {\mathbb N}\), \( x_j\) is not in the closure of the span of \(\{x_1 , \dots, x_{j-1} , x_{j +1} , \ldots\}\). A finitely linearly independent Schauder frame \(\{x_n\}\) is exact if and only if whenever \(\lim_{n\rightarrow\infty} \sum_{i=1}^n a_i x_i = 0\), we have \(\lim_{n\rightarrow\infty} a_n = 0\).