Sums of Frames from the Weyl--Heisenberg Group and Applications to Frame Algorithm

TL;DR

The paper studies sums of Weyl-Heisenberg (Gabor) frames in $L^2(R)$ and derives explicit, verifiable conditions under which finite or infinite sums remain frames with bounds determined by constituent frame bounds and scalar coefficients. It proves a finite-sum sufficiency condition with lower/upper bounds $\left( \sum_{i=1}^k |c_i|^2 A_i - 2 \sum_{i \neq j} |c_i c_j| \sqrt{B_i B_j} \right)$ and $k \sum_{i=1}^k |c_i|^2 B_i$, and shows that the sum of a frame and its dual is always a frame with bounds $A_1+A_2+2$ and $B_1+B_2+2$. Extensions cover sums of images under bounded linear operators with explicit lower bounds $A_1 m_1^2 + A_2 m_2^2 - 2\sqrt{B_1 B_2}\|\Theta_1\|\|\Theta_2\|$, as well as perturbed sums. The authors further demonstrate that the resulting frame bounds can reduce the width $\Delta = (B-A)/(B+A)$, thereby speeding up convergence in the frame algorithm, with concrete examples illustrating improved approximation rates and stability implications.

Abstract

The relationship between the frame bounds of frames (Gabor) for the space $L^2(\mathbb{R})$ with several generators from the Weyl-Heisenberg group and the scalars linked to the sum of frames is examined in this paper. We give sufficient conditions for the finite sum of frames of the space $L^2(\mathbb{R})$ from the Weyl-Heisenberg group, with explicit frame bounds, in terms of frame bounds and scalars involved in the finite sum of frames, to be a frame for $L^2(\mathbb{R})$. It is shown that if a series of square roots of upper frame bounds of countably infinite frames from the Weyl-Heisenberg group is convergent and some lower frame bound majorizes the sum of all other frame bounds, then the infinite sum of frames for $L^2(\mathbb{R})$ space turns out to be a frame for the space $L^2(\mathbb{R})$. We show that the sum of frames from the Weyl-Heisenberg group and its dual frame always constitutes a frame. We provide sufficient conditions for the sum of images of frames under bounded linear operators acting on $L^2(\mathbb{R})$ in terms of lower bounds of their Hilbert adjoint operator to be a frame. The finite sum of frames where frames are perturbed by bounded sequences of scalars is also discussed. As an application of the results, we show that the frame bounds of sums of frames can increase the rate of approximation in the frame algorithm. Our results are true for all types of frames.

Figures (2)

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Theorems & Definitions (10)

• Definition 2.1
• Example 3.3
• Example 3.5
• Remark 3.7
• Example 3.8
• Example 3.10
• Example 4.2
• Example 4.3
• Example 4.4
• Example 4.5