Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products

Frames in a Banach space B were defined as a sequence in its dual space B⁎ in some recent references. We propose to define them as a collection of elements in B by making use of semi-inner products. Classical theory on frames and Riesz bases is generalized under this new perspective. We then aim at establishing the Shannon sampling theorem in Banach spaces. The existence of such expansions in translation invariant reproducing kernel Hilbert and Banach spaces is discussed.