

Response for Referee #1
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1. (p. 1, L17-17) The referee makes a valid, though largely semantic point regarding the use of interpolating in the abstract as it related to GMVR and Runge's phenomenon. However, to avoid any confusion, we have removed discussion of Runge's problem from the text.

2. (p. 1, L32) Thank you. We have added the citation.

3. (p. 1, L44) We have added this reference.

4. (p. 1, L46) We have kept only Leaver's reference at this line. There are indeed many many publications that could be cited here, but Leaver's is one of the earliest and most relevant.

5. (p. 1, L52) We have altered the language here to not mention the topic of completeness.

6. (Third Paragraph) We now no longer constrain this discussion to n=0. The text has been modified to include updated references and now state (perhaps uninformatively) that all phenom models have high frequency regions that are consistent with perturbation theory.

7. (p. 1, L49-50) Thank you. We hope that the current text will avoid any confusion.

8. (p. 1, L52-53) Thank you. We hope that the current text will avoid any confusion.

9. (p. 2, L30) Thank you. We hope that the current text will avoid any confusion.

10. (p. 2, L30) Thank you. We agree that Yang+ should be referenced here.

11. (p. 2, L39) Thank you. We have updated this section to avoid any confusion.

12. (In Eq.6) Thank you. We have clarified the notation.

13. (p. 2, L26-27) Thank you for mentioning the Stone-Weierstrass theorem. We do not wish to reference this theorem in place of the series expansion as the latter is more accessible to a physics audience. We note that our introduction to the discussion is generally applicable given some smooth functoin and an epsilon neighborhood around a point. We intend primarily to motivate a general numtinobial basis perspective of our algorithms from first principles. he text has been modified in an effort to clarify this intention.

14. (Algs. 2 and 5 ...) Thank you. We have clarified that we refer to the set of all multinomial combinations of basis vectors up to a predefined maximum order.

15. (Algs. 2 and 5 ...) Thank you. We have corrected these lines to reference epsilons.

16. (p. 3, L60) The current description of \lambda_{bulk} seeks to communicate the presense of the constant term.

17. (p. 5, L3-4) We have sought to clarify the text to minimize confusion.

18. ( Eq. 20 ) We refer the referee to Eq 21

19. Via Eq. 15, n>=1 for alpha. Here, "implicitely" n plays no real role in the algorithm and therefore need not be stored and increamented.

20. (Alg. 4, step 4) Thank you. We have defined var to be the variance.

21. (Alg. 4, step 4) Thank you. Yes.

22. (p. 8, final line) We have clarified the text.

23. (Fig. 4) We have clarified the text.

24. Code: While we are appreciative of the referee's comments, we point out to the submitted draft only pertains to the GMVP and GMVR algorithms.
