Duality Principles for $$F_a$$-Frame Theory in $$L^2({\mathbb {R}}_+)$$

Abstract

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 $$L^2({{\mathbb {R}}}_+)$$ of square integrable functions on the half real line $${\mathbb {R}}_{+}$$ admits no traditional wavelet or Gabor frame due to $${\mathbb {R}}_{+}$$ being not a group under addition. $$F_{a}$$ -frame theory based on “function-valued inner product” is a new tool for analysis on $$L^2({{\mathbb {R}}}_+)$$ . This paper addresses duality relations for $$F_{a}$$ -frame theory in $$L^2({{\mathbb {R}}}_+)$$ . We introduce the notion of $$F_{a}$$ -R-dual of a given sequence in $$L^2({{\mathbb {R}}}_+)$$ , and obtain some duality principles. Specifically, we prove that a sequence in $$L^2({{\mathbb {R}}}_+)$$ 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 $$F_{a}$$ -Riesz sequence ( $$F_{a}$$ -Bessel sequence, $$F_{a}$$ -Riesz basis, $$F_{a}$$ -frame sequence), and that two sequences in $$L^2({{\mathbb {R}}}_+)$$ form a pair of $$F_{a}$$ -dual frames if and only if their $$F_{a}$$ -R-duals are $$F_{a}$$ -biorthonormal.