# 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:
 [1] P. Binding, L. Boulton, J. Čepička, P. Drábek and P. Girg, Basis properties of eigenfunctions of the p-Laplacian,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 3487.  doi: 10.1090/S0002-9939-06-08001-4.  Google Scholar [2] P. J. Bushell and D. E. Edmunds, Remarks on generalized trigonometric functions,, \emph{Rocky Mountain J. Math.}, 42 (2012), 25.  doi: 10.1216/RMJ-2012-42-1-25.  Google Scholar [3] L. Boulton and G. Lord, Approximation properties of the $q$-sine bases,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, 467 (2011), 2690.  doi: 10.1098/rspa.2010.0486.  Google Scholar [4] B. D. Craven, Stone's theorem and completeness of orthogonal systems,, \emph{J. Austral. Math. Sci.}, 12 (1971), 211.   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 $(|u'| ^{p-2}u')'+f(t,u)=0,u(0)=u(T)=0, p>1$,, \emph{J. Differential Equations}, 80 (1989), 1.  doi: 10.1016/0022-0396(89)90093-4.  Google Scholar [6] O. Došlý, Half-linear differential equations,, \emph{Handbook of differential equations}, (2004), 161.   Google Scholar [7] O. Došlý and P. Řehák, Half-linear Differential Equations,, North-Holland Mathematics Studies, (2005).   Google Scholar [8] P. Drábek, P. Krejčí and P. Takáč, Nonlinear Differential Equations,, Papers from the Seminar on Differential Equations held in Chvalatice, (1998).   Google Scholar [9] P. Drábek and R. Manásevich, On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian,, \emph{Differential Integral Equations}, 12 (1999), 773.   Google Scholar [10] D. E. Edmunds, P. Gurka and J. Lang, Properties of generalized trigonometric functions,, \emph{J. Approx. Theory}, 164 (2012), 47.  doi: 10.1016/j.jat.2011.09.004.  Google Scholar [11] A. Elbert, A half-linear second order differential equation,, \emph{Qualitative theory of differential equations, (1979), 153.   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, (1969).   Google Scholar [13] J. R. Higgins, Completeness and Basis Properties of Sets of Special Functions,, Cambridge Tracts in Mathematics, (1977).   Google Scholar [14] J. Lang and D. E. Edmunds, Eigenvalues, Embeddings and Generalised Trigonometric Functions,, Lecture Notes in Mathematics, (2016).  doi: 10.1007/978-3-642-18429-1.  Google Scholar [15] P. Lindqvist, Some remarkable sine and cosine functions,, \emph{Ricerche Mat.}, 44 (1995), 269.   Google Scholar [16] P. Lindqvist and J. Peetre, $p$-arclength of the $q$-circle,, \emph{The Mathematics Student}, 72 (2003), 139.   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, (1970).   Google Scholar [19] Y. Naito, Uniqueness of positive solutions of quasilinear differential equations,, Differential Integral Equations, 8 (1995), 1813.   Google Scholar [20] F. Qi and Z. Huang, Inequalities of the complete elliptic integrals,, \emph{Tamkang J. Math.}, 29 (1998), 165.   Google Scholar [21] I. Singer, Bases in Banach Spaces. I,, Die Grundlehren der mathematischen Wissenschaften, (1970).   Google Scholar [22] S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with $p$-Laplacian,, \emph{J. Math. Anal. Appl.}, 385 (2012), 24.  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, (1927).  doi: 10.1017/CBO9780511608759.  Google Scholar [24] K. Yosida, Functional Analysis,, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, (1980).   Google Scholar

show all references

##### References:
 [1] P. Binding, L. Boulton, J. Čepička, P. Drábek and P. Girg, Basis properties of eigenfunctions of the p-Laplacian,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 3487.  doi: 10.1090/S0002-9939-06-08001-4.  Google Scholar [2] P. J. Bushell and D. E. Edmunds, Remarks on generalized trigonometric functions,, \emph{Rocky Mountain J. Math.}, 42 (2012), 25.  doi: 10.1216/RMJ-2012-42-1-25.  Google Scholar [3] L. Boulton and G. Lord, Approximation properties of the $q$-sine bases,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, 467 (2011), 2690.  doi: 10.1098/rspa.2010.0486.  Google Scholar [4] B. D. Craven, Stone's theorem and completeness of orthogonal systems,, \emph{J. Austral. Math. Sci.}, 12 (1971), 211.   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 $(|u'| ^{p-2}u')'+f(t,u)=0,u(0)=u(T)=0, p>1$,, \emph{J. Differential Equations}, 80 (1989), 1.  doi: 10.1016/0022-0396(89)90093-4.  Google Scholar [6] O. Došlý, Half-linear differential equations,, \emph{Handbook of differential equations}, (2004), 161.   Google Scholar [7] O. Došlý and P. Řehák, Half-linear Differential Equations,, North-Holland Mathematics Studies, (2005).   Google Scholar [8] P. Drábek, P. Krejčí and P. Takáč, Nonlinear Differential Equations,, Papers from the Seminar on Differential Equations held in Chvalatice, (1998).   Google Scholar [9] P. Drábek and R. Manásevich, On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian,, \emph{Differential Integral Equations}, 12 (1999), 773.   Google Scholar [10] D. E. Edmunds, P. Gurka and J. Lang, Properties of generalized trigonometric functions,, \emph{J. Approx. Theory}, 164 (2012), 47.  doi: 10.1016/j.jat.2011.09.004.  Google Scholar [11] A. Elbert, A half-linear second order differential equation,, \emph{Qualitative theory of differential equations, (1979), 153.   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, (1969).   Google Scholar [13] J. R. Higgins, Completeness and Basis Properties of Sets of Special Functions,, Cambridge Tracts in Mathematics, (1977).   Google Scholar [14] J. Lang and D. E. Edmunds, Eigenvalues, Embeddings and Generalised Trigonometric Functions,, Lecture Notes in Mathematics, (2016).  doi: 10.1007/978-3-642-18429-1.  Google Scholar [15] P. Lindqvist, Some remarkable sine and cosine functions,, \emph{Ricerche Mat.}, 44 (1995), 269.   Google Scholar [16] P. Lindqvist and J. Peetre, $p$-arclength of the $q$-circle,, \emph{The Mathematics Student}, 72 (2003), 139.   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, (1970).   Google Scholar [19] Y. Naito, Uniqueness of positive solutions of quasilinear differential equations,, Differential Integral Equations, 8 (1995), 1813.   Google Scholar [20] F. Qi and Z. Huang, Inequalities of the complete elliptic integrals,, \emph{Tamkang J. Math.}, 29 (1998), 165.   Google Scholar [21] I. Singer, Bases in Banach Spaces. I,, Die Grundlehren der mathematischen Wissenschaften, (1970).   Google Scholar [22] S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with $p$-Laplacian,, \emph{J. Math. Anal. Appl.}, 385 (2012), 24.  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, (1927).  doi: 10.1017/CBO9780511608759.  Google Scholar [24] K. Yosida, Functional Analysis,, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, (1980).   Google Scholar
 [1] Jan Burczak, P. Kaplický. Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2401-2445. doi: 10.3934/cpaa.2016042 [2] Isabelle Déchène. On the security of generalized Jacobian cryptosystems. Advances in Mathematics of Communications, 2007, 1 (4) : 413-426. doi: 10.3934/amc.2007.1.413 [3] Thomas Y. Hou, Pengfei Liu. Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4451-4476. doi: 10.3934/dcds.2016.36.4451 [4] L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure & Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9 [5] Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593 [6] Anna Maria Candela, Addolorata Salvatore. Positive solutions for some generalized $p$–Laplacian type problems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020151 [7] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447 [8] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191 [9] Oliver Junge, Alex Schreiber. Dynamic programming using radial basis functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4439-4453. doi: 10.3934/dcds.2015.35.4439 [10] Sohana Jahan, Hou-Duo Qi. Regularized multidimensional scaling with radial basis functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 543-563. doi: 10.3934/jimo.2016.12.543 [11] Scott W. Hansen, Rajeev Rajaram. Riesz basis property and related results for a Rao-Nakra sandwich beam. Conference Publications, 2005, 2005 (Special) : 365-375. doi: 10.3934/proc.2005.2005.365 [12] Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453 [13] Bo Su. Doubling property of elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 143-147. doi: 10.3934/cpaa.2008.7.143 [14] Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245 [15] Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252 [16] Anurag Jayswal, Ashish Kumar Prasad, Izhar Ahmad. On minimax fractional programming problems involving generalized $(H_p,r)$-invex functions. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1001-1018. doi: 10.3934/jimo.2014.10.1001 [17] Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715 [18] Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371 [19] Shair Ahmad, Alan C. Lazer. On a property of a generalized Kolmogorov population model. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 1-6. doi: 10.3934/dcds.2013.33.1 [20] Sergei A. Avdonin, Boris P. Belinskiy. On the basis properties of the functions arising in the boundary control problem of a string with a variable tension. Conference Publications, 2005, 2005 (Special) : 40-49. doi: 10.3934/proc.2005.2005.40

2018 Impact Factor: 0.925

## Metrics

• PDF downloads (24)
• HTML views (0)
• Cited by (3)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]