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Equivalence of K-Functionals and Modulus of Smoothness Generated by a Generalized Dunkl Operator on the Real Line

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DOI: 10.4236/apm.2015.56035    2,784 Downloads   3,251 Views   Citations

ABSTRACT

This paper is intended to establish the equivalence between K-functionals and modulus of smoothness tied to a Dunkl type operator on the real line.

Conflicts of Interest

The authors declare no conflicts of interest.

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Subaie, R. and Mourou, M. (2015) Equivalence of K-Functionals and Modulus of Smoothness Generated by a Generalized Dunkl Operator on the Real Line. Advances in Pure Mathematics, 5, 367-376. doi: 10.4236/apm.2015.56035.

References

[1] Dunkl, C.F. (1989) Differential-Difference Operators Associated to Reflection Groups. Transactions of the American Mathematical Society, 311, 167-183.
http://dx.doi.org/10.1090/S0002-9947-1989-0951883-8
[2] Dunkl, C.F. (1991) Integral Kernels with Reflection Group Invariance. Canadian Journal of Mathematics, 43, 1213-1227.
http://dx.doi.org/10.4153/CJM-1991-069-8
[3] Dunkl, C.F. (1992) Hankel Transforms Associated to Finite Reflection Groups. Contemporary Mathematics, 138, 128-138.
http://dx.doi.org/10.1090/conm/138/1199124
[4] Kamefuchi, S. and Ohnuki, Y. (1982) Quantum Field Theory and Parastatistics. University of Tokyo Press, Springer-Verlag, Tokyo, Berlin.
[5] Rosenblum, M. (1994) Generalized Hermite Polynomials and the Bose-Like Oscillator Calculus. Operator Theory: Advances and Applications, 73, 369-396.
http://dx.doi.org/10.1007/978-3-0348-8522-5_15
[6] Yang, L.M. (1951) A Note on the Quantum Rule of the Harmonic Oscillator. Physical Review Letters, 84, 788-790.
http://dx.doi.org/10.1103/PhysRev.84.788
[7] Al Sadhan, S.A., Al Subaie, R.F. and Mourou, M.A. (2014) Harmonic Analysis Associated with A First-Order Singular Differential-Difference Operator on the Real Line. Current Advances in Mathematics Research, 1, 23-34.
[8] Al Subaie, R.F. and Mourou, M.A. (2014) Inversion of Two Dunkl Type Intertwining Operators on R Using Generalized Wavelets. Far East Journal of Applied Mathematics, 88, 91-120.
[9] Mourou, M.A. and Trimèche, K. (2003) Transmutation Operators and Paley-Wiener Theorem Associated with a Singular Differential-Difference Operator on the Real Line. Analysis and Applications, 1, 43-69.
http://dx.doi.org/10.1142/S0219530503000090
[10] Trimèche, K. (1981) Transformation intégrale de Weyl et théorème de Paley-Wiener associés à un opérateur différentiel singulier sur . Journal de Mathématiques Pures et Appliquées, 60, 51-98.
[11] Platonov, S.S. (2000) Generalized Bessel Translations and Certain Problems of the Theory of Approximation of Functions in the Metrics of L2,α. I. Trudy Petrozavodskogo Gosudarstvennogo Universiteta, Seriya Matematika, 7, 70-82.
[12] Platonov, S.S. (2001) Generalized Bessel Translations and Certain Problems of the Theory of Approximation of Functions in the Metrics of L2,α. II. Trudy Petrozavodskogo Gosudarstvennogo Universiteta, Seriya Matematika, 8, 1-17.
[13] Potapov, M.K. (1998) Application of the Operator of Generalized Translation in Approximation Theory. Vestnik Moskovskogo Universiteta, Seriya Matematika, Mekhanika, 3, 38-48.
[14] Peetre, J. (1963) A Theory of Interpolation of Normed Spaces. Notes de Universidade de Brasilia, Brasilia.
[15] Berens, H. and Buter, P.L. (1967) Semi-Groups of Operators and Approximation. Grundlehren der mathematischen Wissenschaften, 145, Springer, Berlin.
[16] Belkina, E.S. and Platonov, S.S. (2008) Equivalence of K-Functionnals and Modulus of Smoothness Constructed by Generalized Dunkl Translations. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika, 8, 3-15.
[17] Dai, F. (2003) Some Equivalence Theorems with K-Functionals. Journal of Approximation Theory, 121, 143-157.
http://dx.doi.org/10.1016/S0021-9045(02)00059-X
[18] Ditzian, Z. and Totik, V. (1987) Moduli of Smoothness. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-4778-4
[19] L?fstróm, J. and Peetre, J. (1969) Approximation Theorems Connected with Generalized Translations. Mathematische Annalen, 181, 255-268.
http://dx.doi.org/10.1007/BF01350664

  
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