Approximation of the Inverse Frame Operator and Applications to Gabor Frames

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

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On Discrete Time Wilson Systems

Journal of Mathematics ◽

10.1155/2020/8426897 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Teena Kohli ◽

Suman Panwar ◽

S. K. Kaushik

Keyword(s):

Discrete Time ◽

Dual Pair ◽

Gabor Frames ◽

Sufficient Condition ◽

Zak Transform ◽

Frame Operator

In this paper, we define the discrete time Wilson frame (DTW frame) for l 2 ℤ and discuss some properties of discrete time Wilson frames. Also, we give an interplay between DTW frames and discrete time Gabor frames. Furthermore, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are given. Moreover, the frame operator for the DTW frame is obtained. Finally, we discuss dual pair of frames for discrete time Wilson systems and give a sufficient condition for their existence.

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WEIGHTED AND CONTROLLED FRAMES: MUTUAL RELATIONSHIP AND FIRST NUMERICAL PROPERTIES

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691310003377 ◽

2010 ◽

Vol 08 (01) ◽

pp. 109-132 ◽

Cited By ~ 36

Author(s):

PETER BALAZS ◽

JEAN-PIERRE ANTOINE ◽

ANNA GRYBOŚ

Keyword(s):

Iterative Algorithms ◽

Gabor Frames ◽

Dual Frame ◽

Mutual Relationship ◽

Frame Operator ◽

Numerical Efficiency ◽

Frame Condition ◽

Frame Bound ◽

Special Case ◽

Numerical Advantage

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while offering a numerical advantage in the sense of preconditioning. Next, we investigate weighted frames, in particular their relation to controlled frames. We consider the special case of semi-normalized weights, where the concepts of weighted frames and standard frames are interchangeable. We also make the connection with frame multipliers. Finally, we analyze weighted frames numerically. First, we investigate three possibilities for finding weights in order to tighten a given frame, i.e. decrease the frame bound ratio. Then, we examine Gabor frames and how well the canonical dual of a weighted frame is approximated by the inversely weighted dual frame.

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