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Making and sharing \(K\)-dual frame pairs. (English) Zbl 1461.42022

Summary: The atomic decomposition of signals is one of the most important problems in the frame theory. \(K\)-dual frame pairs may be used to stably reconstruct elements from the range of bounded linear operators on Hilbert spaces. The purpose of this paper is making \(K\)-dual frame pairs and finding common \(K\)-dual Bessel sequence. We present a sufficient condition on operators on \(\mathcal{H}\) which takes a \(K\)-dual frame pairs to other ones; characterize bounded linear operators on \(I^2(\mathcal{J})\) that transform \(K\)-dual frame pairs to other ones; prove that two Bessel sequences can always be extended to a \(K\)-dual frame pair, and that two orthogonal \(K\)-frames have a common \(K\)-dual Bessel sequence under certain conditions; and obtain a sufficient condition which the \(K\)-duals of one \(K\)-frame is contained in the ones of another \(K\)-frames. Abundant examples are also provided to illustrate the generality of the theory.

MSC:

42C15 General harmonic expansions, frames
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
47B38 Linear operators on function spaces (general)
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