عنوان مجله
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International Journal of Wavelets Multiresolution and Information Processing
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چکیده
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Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper,
like the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basi.
Also, we discuss the cases that the operator associated to controlled Gram matrix will
be bounded, invertible, Hilbert–Schmidt or a trace-class operator. Similar to standard
frames, we present an explicit structure for controlled Riesz bases and show that every
(U,C)-controlled Riesz basis {fk}∞
k=1 is in the form {U−1CMek}∞
k=128 , where M is a
bijective operator on H. Furthermore, we propose an equivalent accessible condition to
the sequence {fk}∞
k=1 30 being a (U,C)-controlled Riesz basis.
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