As a completion of the previously published theory of higher-order types of asymptotic variation, we point out some remarks about different approaches to construct such an advanced and demanding theory, because there are various (non-equivalent) approaches to the basic concepts of types of asymptotic variation, i.e., the concepts not involving derivatives. After listing the five known approaches to the concept of regular variation and their mutual relationships, we show that, as far as the higher-order theory is concerned, there is essentially “only one way” to construct such a theory for differentiable functions.
References
[1]
Granata, A. (2016) The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part I: Higher-Order Regular, Smooth and Rapid Variation. Advances in Pure Mathematics, 6, 776-816. https://doi.org/10.4236/apm.2016.612063
[2]
Granata, A. (2016) The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part II: Algebraic Operations and Types of Exponential Variation. Advances in Pure Mathematics, 6, 817-867. https://doi.org/10.4236/apm.2016.612064
[3]
Granata, A. (2019) Complements to the Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Advances in Pure Mathematics, 9, 434-479. https://doi.org/10.4236/apm.2019.95022
[4]
Granata, A. (2021) Operations with Higher-Order Types of Asymptotic Variation: Filling Some Gaps. Advances in Pure Mathematics, 11, 687-716. https://doi.org/10.4236/apm.2021.118046
[5]
Hardy, G.H. (1924) Orders of Infinity. Second Improved Edition, Cambridge University Press.
[6]
Bourbaki, N. (1976) Fonctions d’une variable réelle—Théorie élémentaire (Chap. V). Hermann.
[7]
Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge University Press. https://doi.org/10.1017/cbo9780511721434
[8]
Seneta, E. (1976) Regularly Varying Functions. Springer-Verlag. https://doi.org/10.1007/BFb0079658
[9]
Parameswaran, S. (1961) Partition Functions Whose Logarithms Are Slowly Oscillating. Transactions of the American Mathematical Society, 100, 217-240. https://doi.org/10.1090/s0002-9947-1961-0140498-x
[10]
Granata, A. (2018) Asymptotic Behaviors of Wronskians and Finite Asymptotic Expansions in the Real Domain—Part II: Mixed Scales and Exceptional Cases. International Journal of Advanced Research in Mathematics, 12, 35-68. https://doi.org/10.18052/www.scipress.com/ijarm.12.35
