On p-frames and reconstruction series in separable Banach spaces

It is well known that a frame {g(i)} for a Hilbert space H allows every element f is an element of H to be represented as f = Sigma[f, f(i)] g(i) = Sigma [f, g(i)] f(i) via the frame elements and a dual frame {f(i)}, f(i) is an element of H. For some generalizations of frames to Banach spaces (Banach frames, p-frames), such representations are not always possible. For a given sequence {g(i)} with elements in the dual X* of a Banach space X, we discuss the p-frame condition and validity of series expansions in the form g = Sigma d(i)g(i) for appropriate coefficients {d(i)} and also reconstruction series in the form f = Sigma g(i)(f) f(i), f is an element of X, and g = Sigma g(f(i)) g(i), g is an element of X*, via appropriate sequence {f(i)}, f(i) is an element of X. In particular, we show that a Banach frame w. r. t. l(p) always leads to the desired representations; however, general Banachframes do not.