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An introduction to frames and Riesz bases

An introduction to frames and Riesz bases,Ole Christensen

An introduction to frames and Riesz bases   (Citations: 491)
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Published in 2003.
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    • ...For the Parseval frames, the reconstruction formula holds true 12 37...

    S. Häuseret al. Convex multiclass segmentation with shearlet regularization

    • ...Frame theory has applications to a wide variety of problems in signal processing and much more (see the monographs [12, 15] for a comprehensive view)...

    Bernhard G. Bodmannet al. Fusion Frames and the Restricted Isometry Property

    • ...Frames were introduced by Duffin and Schaeffer in 2, and have been developed very quickly in the past 20 more years, see 1,3,4...

    Zhijing Zhaoet al. Sufficient conditions and stability of wavelet superframes

    • ...This characterization may be viewed as the fusion frame counterpart to Naimark’s theorem [13, 17, 28, 36], where Parseval frames are characterized as frame systems generated by an orthogonal projection of an orthonormal basis from a larger Hilbert space...
    • ...[13, 28, 36]) that there exists a Hilbert space K ⊃ H with an orthonormal basis {˜ eij}i∈I, j∈Ji so that the orthogonal projection P of K onto H satisfies...
    • ...is a Parseval frame for H .B y [13, 28, 36], there exists a Hilbert space K ⊇ H, an orthogonal projection P : K → H, and an orthonormal basis {eij}i∈I, j∈Ji for K so that...

    Robert Calderbanket al. Sparse fusion frames: existence and construction

    • ...It also encompasses a large class of signal representations commonly used in signal processing, including the Gabor and wavelet transforms [16]...
    • ...the uniform time-frequency tiling in Gabor representations [16])...
    • ...Examples include the translation, modulation, and dilatation operators, which are used, respectively, in classical shift-invariant (SI) sampling problems [1], [2], in magnetic resonance imaging (MRI) and Gabor analysis [17], and in wavelet analysis [16], [18], [19]...
    • ...Remark 1: The above theorem is well known for the case in which the sequence is generated by the translation operator given in Example 1 (see, e.g., [16], [19], [24], [34])...
    • ...Consequently, the subset of past sampling functions cannot be a Riesz basis for the whole sampling space . However, it is still a Riesz basis for the past sampling space (see, e.g., [16]) so that causal processing can be pursued in a stable manner...
    • ...and consequently it also cannot be a Riesz basis or a frame for (see, e.g., [16])...
    • ...It remains to show that is a Bessel sequence, Riesz basis, or frame if the spectral density satisfies the conditions of Theorem 6. It is known (see, e.g., [16]) that is a Riesz basis for with Riesz bounds if and only if...

    Tomer Michaeliet al. U-Invariant Sampling: Extrapolation and Causal Interpolation From Gene...

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