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Extremal Functions on Sturm–Liouville Hypergroups

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Abstract

We give an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev spaces associated with a singular second-order differential operator. Next, we come up with some results regarding the multiplier operators for the Sturm–Liouville transform.

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References

  1. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bloom, W.R., Xu, Z.: Fourier multipliers for \(L^p\) on Chébli–Trimèche hypergroups. Proc. Lond. Math. Soc. 80, 643–664 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Castro, L.P., Saitoh, S., Sawano, Y., Simões, A.M.: General inhomogeneous discrete linear partial differential equations with constant coefficients on the whole spaces. Complex Anal. Oper. Theory 6, 307–324 (2012)

    Google Scholar 

  4. Chébli, H.: Théorème de Paley-Wiener associé à un opérateur différentiel singulier sur \((0, \infty )\). J. Math. Pures Appl. 58(1), 1–19 (1979)

    MathSciNet  Google Scholar 

  5. Kimeldorf, G.S., Wahba, G.: Some results on Tchebycheffian spline functions. J. Math. Anal. Appl. 33, 82–95 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  6. Lee, S.: Generalized Sobolev spaces of exponential type. Kangweon-Kyungki Math. J. 8, 73–86 (2000)

    Google Scholar 

  7. Matsuura, T., Saitoh, S., Trong, D.D.: Inversion formulas in heat conduction multidimensional spaces. J. Inv. Ill-Posed Probl. 13, 479–493 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Nessibi, M.M., Rachdi, L.T., Triméche, K.: The local central limit theorem on the product of the Chébli-Trimèche hypergroup and the Euclidean hypergroup \(\mathbb{R}^n\). J. Math. Sci. (Calcutta) 9(2), 109–123 (1998)

    MathSciNet  Google Scholar 

  9. Pahk, D.H., Kang, B.H.: Sobolev spaces in the generalized distribution spaces of Beurling type. Tsukuba J. Math. 15, 325–334 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Saitoh, S.: The Weierstrass transform and an isometry in the heat equation. Appl. Anal. 16, 1–6 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  11. Saitoh, S.: Approximate real inversion formulas of the Gaussian convolution. Appl. Anal. 83, 727–733 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Sawano, Y., Fujiwara, H., Saitoh, S.: Real inversion formulas of the Laplace transform on weighted function spaces. Complex Anal. Oper. Theory 2, 511–521 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Stein, E.M.: Interpolation of linear operators. Trans. Am. Math. Soc. 83, 482–492 (1956)

    Article  MATH  Google Scholar 

  14. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  15. Trimèche, K.: Transformation intégrale de Weyl et théorème de Paley–Wiener associés à un opérateur différentiel singulier sur \((0, \infty )\). J. Math. Pures Appl. 60(1), 51–98 (1981)

    MATH  MathSciNet  Google Scholar 

  16. Xu, Z.: Harmonic analysis on Chébli-Trimèche hypergroups. Ph.D. thesis, Murdoch University, Perth (1994)

  17. Yamada, M., Matsuura, T., Saitoh, S.: Representations of inverse functions by the integral transform with the sign kernel. Frac. Calc. Appl. Anal. 2, 161–168 (2007)

    MathSciNet  Google Scholar 

  18. Zeuner, H.: The central limit theorem for Chébli–Trimèche hypergroups. J. Theor. Probab. 2(1), 51–63 (1989)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Fethi Soltani.

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Communicated by Saburou Saitoh.

F. Soltani partially supported by DGRST project 04/UR/15-02 and CMCU program 10G 1503.

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Soltani, F. Extremal Functions on Sturm–Liouville Hypergroups. Complex Anal. Oper. Theory 8, 311–325 (2014). https://doi.org/10.1007/s11785-013-0303-9

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