In a previous series of papers we established a general theory of finite asymptotic expansions in the real domain for functions* f* of one real variable sufficiently-regular on a deleted neighborhood of a point *x*_{0} ∈ R, a theory based on the use of a uniquely-determined linear differential operator *L* associated to the given asymptotic scale and wherein various sets of asymptotic expansions are characterized by the convergence of improper integrals involving both the operator *L* applied to *f* and certain weight functions constructed by means of Wronskians of the given scale. Very special cases apart, Wronskians have quite complicated expressions and unrecognizable asymptotic behaviors; however in the present work, split in two parts, we highlight some approaches for determining the exact asymptotic behavior of a Wronskian when the involved functions are regularly- or rapidly-varying functions of higher order. This first part contains: (i) some preliminary results on the asymptotic behavior of a determinant whose entries are asymptotically equivalent to the entries of a Vandermonde determinant; (ii) the fundamental results about the asymptotic behaviors of Wronskians involving scales of functions all of which are either regularly (or, more generally, smoothly) varying or rapidly varying of a suitable higher order. A lot of examples and applications to the theory of asymptotic expansions in the real domain are given.

Periodical:

International Journal of Advanced Research in Mathematics (Volume 9)

Pages:

1-33

Citation:

A. Granata "Asymptotic Behaviors of Wronskians and Finite Asymptotic Expansions in the Real Domain - Part I: Scales of Regularly- or Rapidly-Varying Functions", International Journal of Advanced Research in Mathematics, Vol. 9, pp. 1-33, 2017

Online since:

Jun 2017

Authors:

Keywords:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

[1] A. Granata, Analytic theory of finite asymptotic expansions in the real domain. Part I: two-term expansions of differentiable functions, Analysis Mathematica. 37(4) (2011) 245-287.

DOI: https://doi.org/10.1007/s10476-011-0402-7[2] A. Granata, The factorizational theory of finite asymptotic expansions in the real domain: a survey of the main results, Advances in Pure Mathematics. 5 (2015) 1-20.

[3] A. Granata, Analytic theory of finite asymptotic expansions in the real domain. Part II-A: the factorizational theory for Chebyshev asymptotic scales, Advances in Pure Mathematics. 5 (2015) 454-480.

[4] A. Granata, Analytic theory of finite asymptotic expansions in the real domain. Part II-B: solutions of differential inequalities and asymptotic admissibility of standard derivatives, Advances in Pure Mathematics. 5 (2015) 481-502.

[5] A. Granata, Analytic theory of finite asymptotic expansions in the real domain. Part II-C: constructive algorithms for canonical factorizations and a special class of asymptotic scales, Advances in Pure Mathematics. 5 (2015) 503-526.

[6] A. Granata, 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 (2016) 776-816.

[7] A. Granata, 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 (2016) 817-867.

[8] G. Pólya, G. Sz¨ego, Problems and theorems in analysis, Vol. II, Springer-Verlag, Berlin, (1976).

[9] M. Krusemeyer, Why does the Wronskian work?, American Mathematical Monthly. 95 (1988) 46-49.

[10] L. Mina, Formule generali delle derivate successive d'una funzione, espresse mediante quelle della sua inversa, Giornale di Matematiche. 43 (1904) 196-212.

[11] C. Wenchang, The Faà di Bruno formula and determinant identities, Linear and Multilinear Algebra. 54(1) (2006) 1-25.

[12] I. Satake, Linear algebra, Marcel Dekker, Inc., New York, (1975).

[13] A.T. Benjamin, G.P. Dresden, A combinatorial proof of Vandermonde's determinant, American Mathematical Monthly. 114 (2007) 338-341.

[14] N.I. Fel'dman, Approximation of certain transcendental numbers. II: The approximation of certain numbers associated with the Weierstrass ℘-function, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya.15 (1951).

[15] A. Baker, On the periods of the Weierstrass ℘-function, Symposia Mathematica, Vol. IV, INDAM, Rome, 1968/69, Academic Press, London, 1970, pp.155-174.

[16] A. J. Van der Poorten, Some determinants that should be better known, Journal of the Australian Mathematical Society (Series A). 21(03) (1976) 278-288.

[17] D. Zeitlin, A Wronskian, American Mathematical Monthly. 65 (1958) 345-349.

[18] M.S. Knebelman, The Wronskian for linear differential equations, American Mathematical Monthly. 56 (1949) 252-254.

[19] A. Bostan, Ph. Dumas, Wronskians and linear independence, American Mathematical Monthly. 117 (2010) 722-727.

[20] E.N. G¨uichal, A relation between Gram and Wronsky determinants, Linear Algebra and its Applications. 72 (1985) 59-72.

DOI: https://doi.org/10.1016/0024-3795(85)90142-9[21] W.R. Utz, An application of the Wronskian, American Mathematical Monthly. 76 (1969) 56-57.

[22] A. Shiro, T. Hirano, On the proof of a formula of Wronski, Sˆugaku. 29 (1977/78) 364-365.

[23] G.H. Hardy, Orders of infinity. The Infinit¨arcalc¨ul, of Paul du Bois-Reymond, Cambridge the University Press, (1924).

[24] N. Bourbaki, Fonctions d'une variable Réelle-Théorie Élémentaire. Hermann, Paris, (1976).

[25] C. Obimbo, A new method to order functions by asymptotic growth rates, Unpublished. Available: https://www.cs.ubc.ca/wccce/Program03/papers/Obi2.pdf.