[FORTHCOMING] Sums of frames from the Weyl–Heisenberg group and applications to frame algorithm

In this paper, we study frame properties of finite sums of frames from the Weyl-Heisenberg group. We give sufficient conditions for a finite sum of frames of the space $$$L^2(\mathbb{R})$$$ from the Weyl-Heisenberg group, with explicit frame bounds, to be a frame for $$$L^2(\mathbb{R})$$$. These conditions are given in terms of frame bounds and scalars involved in the finite sum of frames. We show that the sum of a frame from the Weyl-Heisenberg group and its dual frame always constitutes a frame. Further, we provide sufficient conditions for the sum of images of frames under bounded linear operators acting on $$$L^2(\mathbb{R})$$$ to be a frame. These are expressed in terms of the lower bounds of their Hilbert-adjoint operator. We also discuss finite sums of frames where the frames are perturbed by bounded sequences of scalars. As an application, we show that the frame bounds of sums of frames can increase the rate of approximation in the frame algorithm.

In this paper, we study frame properties of finite sums of frames from the Weyl-Heisenberg group. We give sufficient conditions for a finite sum of frames of the space $$$L^2(\mathbb{R})$$$ from the Weyl-Heisenberg group, with explicit frame bounds, to be a frame for $$$L^2(\mathbb{R})$$$. These conditions are given in terms of frame bounds and scalars involved in the finite sum of frames. We show that the sum of a frame from the Weyl-Heisenberg group and its dual frame always constitutes a frame. Further, we provide sufficient conditions for the sum of images of frames under bounded linear operators acting on $$$L^2(\mathbb{R})$$$ to be a frame. These are expressed in terms of the lower bounds of their Hilbert-adjoint operator. We also discuss finite sums of frames where the frames are perturbed by bounded sequences of scalars. As an application, we show that the frame bounds of sums of frames can increase the rate of approximation in the frame algorithm.

Frames Gabor frames Weyl-Heisenberg group sum of frames Frame algorithm

Frames Gabor frames Weyl-Heisenberg group sum of frames Frame algorithm