# American Institute of Mathematical Sciences

November  2014, 13(6): 2675-2692. doi: 10.3934/cpaa.2014.13.2675

## The basis property of generalized Jacobian elliptic functions

 1 Department of Mathematical Sciences, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama 337-8570, Japan

Received  October 2013 Revised  February 2014 Published  July 2014

The Jacobian elliptic functions are generalized to functions including the generalized trigonometric functions. The paper deals with the basis property of the sequence of generalized Jacobian elliptic functions in any Lebesgue space. In particular, it is shown that the sequence of the classical Jacobian elliptic functions is a basis in any Lebesgue space if the modulus $k$ satisfies $0 \le k \le 0.99$.
Citation: Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675
##### References:
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##### References:
 [1] P. Binding, L. Boulton, J. Čepička, P. Drábek and P. Girg, Basis properties of eigenfunctions of the p-Laplacian, Proc. Amer. Math. Soc., 134 (2006), 3487-3494. doi: 10.1090/S0002-9939-06-08001-4.  Google Scholar [2] P. J. Bushell and D. E. Edmunds, Remarks on generalized trigonometric functions, Rocky Mountain J. Math., 42 (2012), 25-57. doi: 10.1216/RMJ-2012-42-1-25.  Google Scholar [3] L. Boulton and G. Lord, Approximation properties of the $q$-sine bases, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 2690-2711. doi: 10.1098/rspa.2010.0486.  Google Scholar [4] B. D. Craven, Stone's theorem and completeness of orthogonal systems, J. Austral. Math. Sci., 12 (1971), 211-223.  Google Scholar [5] M. del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'| ^{p-2}u')'+f(t,u)=0,u(0)=u(T)=0, p>1$, J. Differential Equations, 80 (1989), 1-13. doi: 10.1016/0022-0396(89)90093-4.  Google Scholar [6] O. Došlý, Half-linear differential equations, Handbook of differential equations, 161-357, Elsevier/North-Holland, Amsterdam, 2004.  Google Scholar [7] O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005.  Google Scholar [8] P. Drábek, P. Krejčí and P. Takáč, Nonlinear Differential Equations, Papers from the Seminar on Differential Equations held in Chvalatice, June 29-July 3, 1998. Chapman & Hall/CRC Research Notes in Mathematics, 404. Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar [9] P. Drábek and R. Manásevich, On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian, Differential Integral Equations,12 (1999), 773-788.  Google Scholar [10] D. E. Edmunds, P. Gurka and J. Lang, Properties of generalized trigonometric functions, J. Approx. Theory, 164 (2012), 47-56. doi: 10.1016/j.jat.2011.09.004.  Google Scholar [11] A. Elbert, A half-linear second order differential equation, Qualitative theory of differential equations, Vol. I, II (Szeged, 1979), pp. 153-180, Colloq. Math. Soc. Janos Bolyai, 30, North-Holland, Amsterdam-New York, 1981.  Google Scholar [12] I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18 American Mathematical Society, Providence, R.I. 1969  Google Scholar [13] J. R. Higgins, Completeness and Basis Properties of Sets of Special Functions, Cambridge Tracts in Mathematics, Vol. 72. Cambridge University Press, Cambridge-New York-Melbourne, 1977.  Google Scholar [14] J. Lang and D. E. Edmunds, Eigenvalues, Embeddings and Generalised Trigonometric Functions, Lecture Notes in Mathematics, 2016. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18429-1.  Google Scholar [15] P. Lindqvist, Some remarkable sine and cosine functions, Ricerche Mat., 44 (1995), 269-290.  Google Scholar [16] P. Lindqvist and J. Peetre, $p$-arclength of the $q$-circle, The Mathematics Student, 72 (2003), 139-145.  Google Scholar [17] P. Lindqvist and J. Peetre, Comments on Erik Lundberg's 1879 thesis. Especially on the work of Göran Dillner and his influence on Lundberg, Memorie dell'Instituto Lombardo (Classe di Scienze Matem. Nat.) 31, 2004. Google Scholar [18] D. S. Mitrinović, Analytic inequalities, In cooperation with P. M. Vasić, Die Grundlehren der mathematischen Wissenschaften, Band 165 Springer-Verlag, New York-Berlin, 1970.  Google Scholar [19] Y. Naito, Uniqueness of positive solutions of quasilinear differential equations, Differential Integral Equations, 8 (1995), 1813-1822.  Google Scholar [20] F. Qi and Z. Huang, Inequalities of the complete elliptic integrals, Tamkang J. Math., 29 (1998), 165-169.  Google Scholar [21] I. Singer, Bases in Banach Spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 154. Springer-Verlag, New York-Berlin, 1970.  Google Scholar [22] S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with $p$-Laplacian, J. Math. Anal. Appl., 385 (2012), 24-35. doi: 10.1016/j.jmaa.2011.06.063.  Google Scholar [23] E. T. Whittaker and G. N. Watson, A course of modern analysis, An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Reprint of the fourth (1927) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511608759.  Google Scholar [24] K. Yosida, Functional Analysis, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Google Scholar
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