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.

Casazza等人首先介绍了一般希尔伯特空间中的R-对偶概念。（J Fourier Anal Appl 10：383–408，2004），目的是在Gabor分析中获得对偶原理的通用版本。另一方面，半实线\（{\ mathbb {R}} _ {+} \上的平方可积函数的空间\（L ^ 2（{{\ mathbb {R}} __ +）\））由于\（{\ mathbb {R}} _ {+} \）不是加法的群组，因此不接受传统的小波或Gabor框架。\（F_ {a} \）-基于“函数值内积”的框架理论是一种用于分析\（L ^ 2（{{\ mathbb {R}}} _ +）\）的新工具。本文针对\（L ^ 2（{{\ mathbb {R}}} _ +）\）中\（F_ {a} \）-框架理论的对偶关系。我们引入\（L ^ 2（{{\ mathbb {R}} _ +）\）\）中给定序列\（F_ {a} \）- R-dual的概念，并获得一些对偶原理。具体来说，我们证明\（L ^ 2（{{\ mathbb {R}}} _ +）\）中的序列是\（F_ {a} \）- frame（\（F_ {a} \） -贝塞尔序列，\（F_ {a} \）- Riesz基，\（F_ {a} \）-帧序列，当且仅当其\（F_ {a} \）- R-对偶是\（F_ { a} \）- Riesz序列（\（F_ {a} \）-贝塞尔序列，\（F_ {a} \）- Riesz基，\（F_ {a} \）-帧序列），以及\（L ^ 2（{{\\ mathbb {R}}} _ +）\）当且仅当它们的\（F_ {a} \） - R-对偶是\（F_ {a} \） -双正交时，才形成一对\（F_ {a} \）-对偶框架。