On p-frames and reconstruction series in separable Banach spaces

Abstract

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