Duality Principles for Fa-Frame Theory in L2(R+)

The notion of R-dual in general Hilbert spaces was first introduced by Casazza et al. (J Fourier Anal Appl 10:383-408, 2004), with the motivation to obtain a general version of the duality principle in Gabor analysis. On the other hand, the space L-2(R+) of square integrable functions on the half real line R+ admits no traditional wavelet or Gabor frame due to R+ being not a group under addition. Fa-frame theory based on "function-valued inner product" is a new tool for analysis on L-2(R+). This paper addresses duality relations for Fa-frame theory in L-2(R+). We introduce the notion of Fa-R-dual of a given sequence in L-2(R+), and obtain some duality principles. Specifically, we prove that a sequence in L-2(R+) is an F-a-frame (F-a-Bessel sequence, F-a-Riesz basis, F-a-frame sequence) if and only if its F-a-R-dual is an FaRiesz sequence (F-a-Bessel sequence, F-a-Riesz basis, Fa-frame sequence), and that two sequences in L-2(R+) form a pair of Fa-dual frames if and only if their Fa-R-duals are F-a-biorthonormal.