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Duality principles for \(F_a \)-frame theory in \(L^2 (\mathbb{R}_+ )\). (English) Zbl 1472.42047

Authors’ abstract: The notion of R-dual in general Hilbert spaces was first introduced by P. G. Casazza et al. [J. Fourier Anal. Appl. 10, No. 4, 383–408 (2004; Zbl 1058.42020)], 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.

MSC:

42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A30 Approximation by other special function classes

Citations:

Zbl 1058.42020
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References:

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