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arxiv: 2306.09493 · v2 · submitted 2023-06-15 · 🧮 math.FA

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

Divya Jindal , Jyoti , Lalit Kumar Vashisht This is my paper

Pith reviewed 2026-05-24 08:38 UTC · model grok-4.3

classification 🧮 math.FA
keywords Weyl-Heisenberg groupframesGabor framesframe sumsframe boundsframe algorithmL2(R)
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The pith

Sufficient conditions on frame bounds make finite and infinite sums of Weyl-Heisenberg frames into frames for L2(R).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives conditions under which the sum of frames generated by the Weyl-Heisenberg group for L2(R) is itself a frame. For finite sums, explicit frame bounds are given in terms of the original bounds and the summing scalars. For infinite sums, convergence of the square roots of upper bounds together with one lower bound dominating the others ensures the sum is a frame. Additional results show that a frame plus its dual is always a frame, and similar conditions apply to operator images and scalar perturbations. These summed frames improve the convergence rate of the frame algorithm.

Core claim

If a collection of Weyl-Heisenberg frames for L2(R) with bounds A_i and B_i satisfy that the sum of sqrt(B_i) converges and the smallest A_j majorizes the sum of the remaining B_i, then their infinite sum is a frame for L2(R) with positive lower and upper bounds. Analogous explicit conditions hold for finite sums, and the sum of any frame with its dual frame is always a frame.

What carries the argument

Majorization conditions relating the lower and upper frame bounds of the individual Weyl-Heisenberg frames in the sum.

If this is right

  • The frame algorithm achieves faster approximation rates when applied to summed frames with improved lower bounds.
  • A frame and its dual always sum to a frame for L2(R).
  • Bounded linear operators applied to frames yield sums that are frames if the adjoints satisfy lower bound conditions.
  • Scalar perturbations of frames by bounded sequences preserve the frame property under the given conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These majorization rules could be used to construct frames with very large lower bounds by adding sufficiently many individual frames.
  • The results suggest that similar summation techniques might apply to frames in other Hilbert spaces or generated by different groups.
  • Practical implementations of frame-based signal processing could benefit from pre-summing frames to optimize algorithm performance.

Load-bearing premise

The starting collections must each be frames generated by the Weyl-Heisenberg group with explicitly known positive lower and upper frame bounds.

What would settle it

Construct or compute a specific infinite collection of Weyl-Heisenberg frames where sum sqrt(B_i) converges and A_1 exceeds sum of other B_i, yet the summed system fails the lower frame bound inequality for some test function in L2(R).

Figures

Figures reproduced from arXiv: 2306.09493 by Divya Jindal, Jyoti, Lalit Kumar Vashisht.

Figure 1
Figure 1. Figure 1: The following example provides an application of Theorem 3.4 in the frame algorithm. Example 4.3. Consider the frames F : {fk} 5 k=1 = n √ 1 3 , 0, 0  ,  0, √ 1 3 , 0  ,  0, 0, √ 1 3  , √ 2, 0, 0  ,  0, 0, √ 2 o for the complex Hilbert space H = C 3 with frame bounds A1 = 1 3 and B1 = 7 3 . Its dual frame is given by G : {gk} 5 k=1 = n√ 3 2 , 0, 0  ,  0, √ 3, 0  ,  0, 0, √ 3 2  ,  1 2 √ 2 … view at source ↗
Figure 2
Figure 2. Figure 2: In the next example, we show that the frame bounds of sum of images of two frames can increase the rate of approximation in the frame algorithm. This illustrates Theorem 3.6. Example 4.4. Consider the frame F := {fk} 3 k=1 = {( √ 6, √ 6),(0, 2),(2, 0)} for C 2 given in Exam￾ple 4.2 with frame bounds A1 = 4 and B1 = 16. Now the collection of vectors G := {gk} 3 k=1 = {(2, 0),(0, √ 2),(0, √ 2)} is a tight fr… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies sums of frames generated by the Weyl-Heisenberg group in L²(ℝ). It claims sufficient conditions (with explicit bounds) under which finite sums of such frames remain frames, that the sum of a frame and its dual is always a frame, that images under bounded operators preserve the frame property under adjoint lower-bound conditions, and that scalar perturbations preserve frames. For countably infinite sums it asserts that convergence of ∑√B_i together with the majorization “some lower frame bound majorizes the sum of all other frame bounds” yields a frame. Applications to convergence rates of the frame algorithm are given. The claims are stated to hold for all types of frames.

Significance. Correct criteria for when sums of Weyl-Heisenberg frames remain frames would be useful for constructing new frames and for controlling approximation speed in the frame algorithm. The explicit-bound claims and the dual-frame result are potentially valuable if the derivations are valid; the infinite-sum statement is the most ambitious part of the work.

major comments (1)
  1. [abstract; theorem on infinite sums (likely §3)] Abstract and the theorem on infinite sums: the claimed sufficient condition (“some lower frame bound majorizes the sum of all other frame bounds,” i.e., A_k > ∑_{j≠k} B_j) does not guarantee a positive lower frame bound for the summed collection. The natural estimate via the triangle inequality on the analysis operators yields only the weaker requirement √A_k > ∑_{j≠k} √B_j. The counter-example A=4, B₂=B₃=1 satisfies the paper’s majorization yet produces a zero lower-bound estimate; explicit cancellation can destroy the frame property. Unless a stronger estimate is derived later in the manuscript, the stated condition is not sufficient for the central claim.
minor comments (2)
  1. [abstract] The final sentence of the abstract (“Our results are true for all types of frames”) is broader than the hypotheses actually used (Weyl-Heisenberg frames with known A_i, B_i).
  2. [section introducing infinite sums] Notation for the summed collection {∑_i f_n^i}_n and its analysis operator should be introduced explicitly before the infinite-sum theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the infinite-sum theorem. We address the concern directly below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [abstract; theorem on infinite sums (likely §3)] Abstract and the theorem on infinite sums: the claimed sufficient condition (“some lower frame bound majorizes the sum of all other frame bounds,” i.e., A_k > ∑_{j≠k} B_j) does not guarantee a positive lower frame bound for the summed collection. The natural estimate via the triangle inequality on the analysis operators yields only the weaker requirement √A_k > ∑_{j≠k} √B_j. The counter-example A=4, B₂=B₃=1 satisfies the paper’s majorization yet produces a zero lower-bound estimate; explicit cancellation can destroy the frame property. Unless a stronger estimate is derived later in the manuscript, the stated condition is not sufficient for the central claim.

    Authors: We agree that the condition A_k > ∑_{j≠k} B_j as stated is not sufficient to guarantee a positive lower frame bound. The analysis operator of the summed collection satisfies T = ∑ T_i, so the reverse triangle inequality yields the estimate ||Tx|| ≥ ||T_k x|| − ∑_{j≠k} ||T_j x|| ≥ (√A_k − ∑_{j≠k} √B_j) ||x||. Consequently the correct sufficient condition is √A_k > ∑_{j≠k} √B_j (together with the already-assumed convergence of ∑ √B_i). The counter-example correctly illustrates that the original majorization can fail to produce a frame. No stronger estimate appears later in the manuscript that would rescue the stated condition. We will therefore revise both the abstract and the theorem on infinite sums (Section 3) to replace the incorrect majorization with the triangle-inequality version √A_k > ∑_{j≠k} √B_j, while retaining the convergence hypothesis on the square-root upper bounds. The revised statement will also note that the argument is valid for arbitrary frames in a Hilbert space, not only Weyl–Heisenberg frames. revision: yes

Circularity Check

0 steps flagged

No circularity: conditions stated directly in terms of input frame bounds

full rationale

The paper derives sufficient conditions for finite and infinite sums of Weyl-Heisenberg frames to remain frames, expressing the resulting bounds explicitly in terms of the given A_i and B_i of the individual frames (abstract and stated claims). No step reduces the target frame property to a quantity defined by fitting the same data, no self-citation is load-bearing for the central claims, and no ansatz or uniqueness result is smuggled in. The derivation chain is therefore self-contained against the supplied inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard definition of a frame in a Hilbert space and the algebraic properties of the Weyl-Heisenberg group action; no numerical constants are fitted and no new entities are postulated.

axioms (2)
  • standard math L2(R) is a Hilbert space and the Weyl-Heisenberg group acts by unitary time-frequency shifts that preserve the inner-product structure.
    Invoked throughout the statements on frame bounds for sums.
  • standard math A collection {g_i} is a frame if there exist A,B > 0 such that A||f||^2 <= sum |<f,g_i>|^2 <= B||f||^2 for all f.
    The entire paper is built on this definition and the associated frame operator.

pith-pipeline@v0.9.0 · 5812 in / 1631 out tokens · 60345 ms · 2026-05-24T08:38:34.586197+00:00 · methodology

discussion (0)

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Reference graph

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