Some Upper Bounds for RKHS Approximation by Bessel Functions

Round 1

Reviewer 1 Report

Paper presents reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory. To solve the analysed problem the Fourier-Bessel series, Fourier-Bessel transforms and semigroup of operators are used. Authors presents some notions and results of Fourier- 
Bessel series and Fourier-Bessel transforms.  Two kinds of RKHSs 
were constructed. The best RKHS approximation problem arising from learning theory was presented. Some K-functionals and moduli of smoothness associated with Fourier-Bessel series were provided. Some upper bounds for the best 
approximation were shown.  The further analysis for the results of the present paper are given in final conclusion. The manuscript contains new elements in relation to the current state of knowledge in the field of the subject. There are original elements, such as: strong equivalent relation.

The manuscript contains new elements in relation to the current state of knowledge in the field of the subject. There are original elements, such as: strong equivalent relation. The article may be published in a journal after clarifying certain doubts:
Abstract is too general. It would need to be redrafted into a more detailed form.
There are no conclusions from the thesis and no indication of the applications of the developed formulas and the possibility of their implementation.
Detailed comments:

Lines 55-58: In Section 3 some K-functionals and moduli of smoothness associated with Fourier
Bessel series are provided, with which some upper bounds for the best approximation
are shown; Also some K-functionals and moduli of smoothness associated with Fourier
Bessel transforms are provided in Section 4, with which some upper bounds for the best
approximation are shown as well; - It is unclear and it sounds almost the same. These sentences should be reworded.
Line 233: To increase the smoothness for the hypothesis space Besov spaces are chosen.
Why Besov spaces? Can other solutions be used?
Line 240: Moreover, it is the first time that a Jackson inequality
is established to describe the decay. Explain the meaning and application of Jackson inequality.

The manuscript may be accepted for publication after minor revision.

Author Response

See the revision report inthe attachment.

Reviewer 2 Report

The paper is concerned with the reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory. The study shows some K-functionals and moduli of smoothness with respect to RKHSs along with their definitions and their equivalences.

Kindly find below my remarks:

- The language is well-written and the paper is jargon wise correct.

- By emphasizing the advantage of this work, the future directions’ addition would further improve the paper.

- The study can be compared with other works and the novel aspects of the study can be stated better.

- The novel aspects of the paper can be enhanced if the study is presented along with examples. If possible, the authors can add the results on the coordinate plane.

- The abstract can also be expanded as well by making the aim of the study more evident.

- English editing can be done, for example, the grammar errors can be corrected. “best approximation are provided”, in that sentence, “are” should be corrected as “is”

Overall, the paper can be improved after including the aspects mentioned in the comments above.  

Yours faithfully,

Author Response

Pleasesee the revisionreport in the attachment.

Author Response File: Author Response.docx