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Some results on g-frames in Hilbert spaces. (English) Zbl 1242.41030
Summary: We show that every $g$-frame for a Hilbert space ${\cal H}$ can be represented as a linear combination of two $g$-orthonormal bases if and only if it is a $g$-Riesz basis. We also show that every $g$-frame can be written as a sum of two tight $g$-frames with $g$-frame bounds one or a sum of a $g$-orthonormal basis and a $g$-Riesz basis for ${\cal H}$. We further give necessary and sufficient conditions on $g$-Bessel sequences $\{\Lambda_i\in{\cal L}({\cal H},{\cal H}_i):i\in J\}$ and $\{\Gamma_i\in{\cal L}({\cal H},{\cal H}_i):i\in J\}$ and operators $L_1$, $L_2$ on ${\cal H}$ so that $\{\Lambda_iL_1+\Gamma_iL_2:i\in J\}$ is a $g$-frame for ${\cal H}$. We next show that a $g$-frame can be added to any of its canonical dual $g$-frame to yield a new $g$-frame.

41A58Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
42C15General harmonic expansions, frames
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