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 and 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, Proc. Amer. Math. Soc., 134 (2006), 3487-3494. doi: 10.1090/S0002-9939-06-08001-4.

[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.

[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.

[4]

B. D. Craven, Stone's theorem and completeness of orthogonal systems, J. Austral. Math. Sci., 12 (1971), 211-223.

[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.

[6]

O. Došlý, Half-linear differential equations, Handbook of differential equations, 161-357, Elsevier/North-Holland, Amsterdam, 2004.

[7]

O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005.

[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.

[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.

[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.

[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.

[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

[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.

[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.

[15]

P. Lindqvist, Some remarkable sine and cosine functions, Ricerche Mat., 44 (1995), 269-290.

[16]

P. Lindqvist and J. Peetre, $p$-arclength of the $q$-circle, The Mathematics Student, 72 (2003), 139-145.

[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.

[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.

[19]

Y. Naito, Uniqueness of positive solutions of quasilinear differential equations, Differential Integral Equations, 8 (1995), 1813-1822.

[20]

F. Qi and Z. Huang, Inequalities of the complete elliptic integrals, Tamkang J. Math., 29 (1998), 165-169.

[21]

I. Singer, Bases in Banach Spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 154. Springer-Verlag, New York-Berlin, 1970.

[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.

[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.

[24]

K. Yosida, Functional Analysis, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.

show all references

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.

[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.

[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.

[4]

B. D. Craven, Stone's theorem and completeness of orthogonal systems, J. Austral. Math. Sci., 12 (1971), 211-223.

[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.

[6]

O. Došlý, Half-linear differential equations, Handbook of differential equations, 161-357, Elsevier/North-Holland, Amsterdam, 2004.

[7]

O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005.

[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.

[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.

[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.

[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.

[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

[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.

[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.

[15]

P. Lindqvist, Some remarkable sine and cosine functions, Ricerche Mat., 44 (1995), 269-290.

[16]

P. Lindqvist and J. Peetre, $p$-arclength of the $q$-circle, The Mathematics Student, 72 (2003), 139-145.

[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.

[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.

[19]

Y. Naito, Uniqueness of positive solutions of quasilinear differential equations, Differential Integral Equations, 8 (1995), 1813-1822.

[20]

F. Qi and Z. Huang, Inequalities of the complete elliptic integrals, Tamkang J. Math., 29 (1998), 165-169.

[21]

I. Singer, Bases in Banach Spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 154. Springer-Verlag, New York-Berlin, 1970.

[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.

[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.

[24]

K. Yosida, Functional Analysis, Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.

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