The basis property of generalized Jacobian elliptic functions

Shingo Takeuchi

doi:10.3934/cpaa.2014.13.2675

Article Contents

2014, Volume 13, Issue 6: 2675-2692. Doi: 10.3934/cpaa.2014.13.2675

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The basis property of generalized Jacobian elliptic functions

Shingo Takeuchi^1,

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

Received: September 30, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

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 $0 \le k \le 0.99$.

Keywords:

Basis property,
generalized Jacobian elliptic functions,
$p$-Laplacian.

Mathematics Subject Classification: Primary: 34L30, 33E05; Secondary: 34L10, 42A65, 41A30.

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