Increasing Stability in the Inverse Source Problem with an Interval (K₁, K₂) of Frequencies

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

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Comments for author File: Comments.pdf

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Comments on “mathematics-3459231 - Increasing stability in the inverse source problem with an interval (K₁, K₂) of frequencies”

The author investigates the rising stability in the inverse source problem inside a frequency interval (𝐾₁,𝐾₂). Lipschitz-type data discrepancy and the frequency tail of the source function reflect the results showing that bigger wave number intervals improve instability. Particularly, the stability improves as 𝐾₂ increases or drops, so stressing the important part the frequency range plays. The method uses the Fourier transform to provide precise bounds for analytic continuation, therefore offering a strong framework for stability analysis. These findings advance the theoretical knowledge of inverse source problems and provide guidance for useful applications where precise reconstruction of sources is absolutely vital. I have some remarks to edit the work.

• Make the introduction stronger.
• Add more details to the discussion:
  • Add practical advantages are provided by achieving increasing stability in inverse problems?
  • Add how might these findings be useful in other types of inverse problems?
• Make all references follow the same style.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

1.       The phrase "where the latter decreases as the the frequency K2 increases" in the abstract contains a duplicated "the", which should be corrected for clarity.

2.       In section 1.1, the sentence "without loss of generality, we assume K1,K2 > 0" lacks a space after the comma, making it less readable.

3.       Equation (1.3) introduces the Sommerfeld radiation condition, but this notation appears without prior explanation. A brief introduction or reference would help readers unfamiliar with this concept.

4.       In the introduction, while the motivation for the problem is well-presented, the transition between the Helmholtz equation and the inverse problem formulation could be smoother for better logical flow.

5.       Section 1.3 cites several references in a general manner, such as "See also [3, 17, 18, 13, 14, 21]", without explicitly stating their relevance. Consider integrating them directly into the discussion.

6.       In Theorem 2.1, the notation ​  is used without a prior clear definition of the norm. It would be useful to either define it or provide a reminder.

7.       The proof of Theorem 2.1 refers to Lemma 3.3, but it does not give an intuitive explanation of how this lemma contributes to the proof. A brief clarification before applying it would improve readability.

8.       In section 3, integral notations, such as in equation (3.11), sometimes appear without clearly stated limits. Consistently formatting them will improve comprehension.

 

 

Author Response

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Author Response File: Author Response.pdf