Review Reports
by
- Fatih Temizsu
Reviewer 1: Anonymous Reviewer 2: Anonymous Reviewer 3: Ahmed Sani Reviewer 4: Anonymous
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe results of this manuscript are novel and quite interesting. However, some revisions should be made before publication. The author should revise the manuscript according to my suggestions in the attachment.
Comments for author File: 
Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe manuscript introduces a statistical boundedness index η(x) and the associated family of sequence spaces {Xr} within the framework of statistical boundedness. The exposition is well organized, with proofs based on elementary counting arguments, and the topic fits the scope of the journal. My main comments are as follows:
- Novelty is limited. η(x) is essentially an equivalent formulation (by taking the infimum) of statistical boundedness of order ρ introduced in [17]; the solidness, the sequence-algebra property, and the duality computations of Xr are standard results in the theory of sequence spaces. I suggest clarifying the incremental contribution relative to the existing family {Sbρ} and adding a concrete application (e.g., Korovkin-type approximation); otherwise, the value of the work may be questioned.
- Some proofs are incomplete. In the proof of Theorem 4(2), the assumption r+ε≤1 excludes the case r=1, which needs to be treated separately, namely the algebraic closure of X1=Sb. Proposition 2 is stated only for real scalars, which is inconsistent with the complex-valued setting of the paper. In Theorem 4(3), the inequality C{k≤n:|yk|>L}≥n/2−1 does not hold for every L≥0 as stated, since the counting actually depends on L; the argument should be formulated as a lower bound in the limit.
- 3. The keyword “numerical index” may easily be confused with the numerical index of Banach spaces; it should be replaced by “statistical boundedness index”.
Overall, the results are correct and the exposition is sound, but the depth of novelty is limited. I recommend publication after the above revisions have been addressed.
Comments for author File: Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsAll results are quite good, mainly theorem 5 which asserts that for $r\in [0,1]$, the three spaces $X_r^{\alpha}=X_r^{\beta}=X_r^{\gamma}$ coincide with the space of all all finitely nonzero sequences. This is a remarkable result, since it asserts that, in this case the spaces $X_r$ do not coincide with the space of all polynomial functions (polynomials). It will certainly be more interesting if the authors investigate this possibility to characterize $\mathbb R[X]$.
A finalremark (question)
In chain $l^\infty =X_0\subset X_{r_1}\subset X_{r_2}\subset X_1=Sb$, what is the nature of the interpolated spaces between $ X_{r_1}$ and $ X_{r_2}$. Are there other spaces different from $X_r$ with $ {r_1}<r< X_{r_2}$?
Comments for author File: Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 4 Report
Comments and Suggestions for AuthorsSee attached comments.
Comments for author File: Comments.pdf
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 4 Report
Comments and Suggestions for AuthorsI am satisfied that the revised version addresses the concerns raised in my original report, and I have noted further improvements to the manuscript. I have no remaining objections to recommending it for publication.