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May  2015, 14(3): 743-757. doi: 10.3934/cpaa.2015.14.743

General types of spherical mean operators and $K$-functionals of fractional orders

1. 

Departamento de Matemática, Universidade de São Paulo (ICMC - USP), São Carlos, SP 13560-970, Brazil

2. 

Department of Mathematics, Missouri State University, Springfield, MO 65804, United States

Received  March 2014 Revised  September 2014 Published  March 2015

We design a general type of spherical mean operators and employ them to approximate $L_p$ class functions. We show that optimal orders of approximation are achieved via appropriately defined K-functionals of fractional orders.
Citation: Thaís Jordão, Xingping Sun. General types of spherical mean operators and $K$-functionals of fractional orders. Communications on Pure & Applied Analysis, 2015, 14 (3) : 743-757. doi: 10.3934/cpaa.2015.14.743
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,, National Bureau of Standards Applied Mathematics Series 55, (1964).

[2]

G. E. Andrews, R. Askey and R. Roy, Special Functions,, Encyclopedia of Mathematics and Its Applications, (1999). doi: 10.1017/CBO9781107325937.

[3]

W. N. Bailey, Generalized Hypergeometric Series,, Cambridge Tracts in Mathematics and Mathematical Physics, (1964).

[4]

E. Belinsky, F. Dai and Z. Ditzian, Multivariate approximating averages,, \emph{J. Approx. Theory 125}, 1 (2003), 85. doi: 10.1016/j.jat.2003.09.005.

[5]

W. O. Bray, Growth and integrability of Fourier transforms on Euclidean spaces,, to appear in \emph{J. Fourier Analysis and Applications}, (). doi: 10.1007/s00041-014-9354-1.

[6]

W. O. Bray and M. A. Pinsky, Growth properties of Fourier transforms via moduli of continuity,, \emph{J. Funct. Anal. 255}, 9 (2008), 2265. doi: 10.1016/j.jfa.2008.06.017.

[7]

D. Chen, V. A. Menegatto and X. Sun, A necessary and sufficient condition for strictly positive definite functions on spheres,, \emph{Proceedings of the American Mathematical Society 131}, 9 (2003), 2733. doi: 10.1090/S0002-9939-03-06730-3.

[8]

F. Dai and Z. Ditzian, Combinations of multivariate averages,, \emph{J. Approx. Theory 131}, 2 (2004), 268. doi: 10.1016/j.jat.2004.10.003.

[9]

Z. Ditzian, Fractional derivatives and best approximation,, \emph{Acta Math. Hungar. 81}, 4 (1998), 323. doi: 10.1023/A:1006554907440.

[10]

Z. Ditzian, Smoothness of a function and the growth of its Fourier transform or its Fourier coefficients,, \emph{J. Approx. Theory 162}, 5 (2010), 980. doi: 10.1016/j.jat.2009.11.003.

[11]

Z. Ditzian, Relating smoothness to expressions involving Fourier coefficient or to a Fourier transform,, \emph{J. Approx. Theory 164}, 10 (2012), 1369. doi: 10.1016/j.jat.2012.05.004.

[12]

R. E. Edwards, Fourier Series: A Modern Introduction, Vol. II,, Holt, (1967).

[13]

D. Gioev, Moduli of continuity and average decay of Fourier transforms: two sided estimates,, in \emph{Integrable Systems and Random Matrices} (Cont. Math. 458), (2008), 377. doi: 10.1090/conm/458/08948.

[14]

D. Gorbachev and S. Tikhonov, Moduli of smoothness and growth properties of Fourier transforms: two sided estimates,, \emph{J. Approx. Theory 164}, 9 (2012), 1283. doi: 10.1016/j.jat.2012.05.017.

[15]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products,, 6$^{nd}$ edition, (2000).

[16]

F. J. Narcowich, X. Sun and J. D. Ward, Approximation power of RBFs and their associated SBFs: a connection,, \emph{Adv. Comput. Math. 27}, 1 (2007), 107. doi: 10.1007/s10444-005-7506-1.

[17]

I. J. Schoenberg, Metric spaces and completely monotone functions,, \emph{The Annals of Mathematics 39}, 4 (1938), 811. doi: 10.2307/1968466.

[18]

I. J. Schoenberg, Positive definite functions on spheres,, \emph{Duke Math. J.}, 9 (1988), 96.

[19]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2$^{nd}$ edition, (1994). doi: 10.1007/978-1-4612-0873-0.

[20]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).

[21]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces,, Princeton Mathematical Series, (1971).

[22]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals,, 3$^{nd}$ edition, (1986).

[23]

C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations,, \emph{Discrete Continuous Dynam. Systems - B}, 4 (2004), 1065. doi: 10.3934/dcdsb.2004.4.1065.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,, National Bureau of Standards Applied Mathematics Series 55, (1964).

[2]

G. E. Andrews, R. Askey and R. Roy, Special Functions,, Encyclopedia of Mathematics and Its Applications, (1999). doi: 10.1017/CBO9781107325937.

[3]

W. N. Bailey, Generalized Hypergeometric Series,, Cambridge Tracts in Mathematics and Mathematical Physics, (1964).

[4]

E. Belinsky, F. Dai and Z. Ditzian, Multivariate approximating averages,, \emph{J. Approx. Theory 125}, 1 (2003), 85. doi: 10.1016/j.jat.2003.09.005.

[5]

W. O. Bray, Growth and integrability of Fourier transforms on Euclidean spaces,, to appear in \emph{J. Fourier Analysis and Applications}, (). doi: 10.1007/s00041-014-9354-1.

[6]

W. O. Bray and M. A. Pinsky, Growth properties of Fourier transforms via moduli of continuity,, \emph{J. Funct. Anal. 255}, 9 (2008), 2265. doi: 10.1016/j.jfa.2008.06.017.

[7]

D. Chen, V. A. Menegatto and X. Sun, A necessary and sufficient condition for strictly positive definite functions on spheres,, \emph{Proceedings of the American Mathematical Society 131}, 9 (2003), 2733. doi: 10.1090/S0002-9939-03-06730-3.

[8]

F. Dai and Z. Ditzian, Combinations of multivariate averages,, \emph{J. Approx. Theory 131}, 2 (2004), 268. doi: 10.1016/j.jat.2004.10.003.

[9]

Z. Ditzian, Fractional derivatives and best approximation,, \emph{Acta Math. Hungar. 81}, 4 (1998), 323. doi: 10.1023/A:1006554907440.

[10]

Z. Ditzian, Smoothness of a function and the growth of its Fourier transform or its Fourier coefficients,, \emph{J. Approx. Theory 162}, 5 (2010), 980. doi: 10.1016/j.jat.2009.11.003.

[11]

Z. Ditzian, Relating smoothness to expressions involving Fourier coefficient or to a Fourier transform,, \emph{J. Approx. Theory 164}, 10 (2012), 1369. doi: 10.1016/j.jat.2012.05.004.

[12]

R. E. Edwards, Fourier Series: A Modern Introduction, Vol. II,, Holt, (1967).

[13]

D. Gioev, Moduli of continuity and average decay of Fourier transforms: two sided estimates,, in \emph{Integrable Systems and Random Matrices} (Cont. Math. 458), (2008), 377. doi: 10.1090/conm/458/08948.

[14]

D. Gorbachev and S. Tikhonov, Moduli of smoothness and growth properties of Fourier transforms: two sided estimates,, \emph{J. Approx. Theory 164}, 9 (2012), 1283. doi: 10.1016/j.jat.2012.05.017.

[15]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products,, 6$^{nd}$ edition, (2000).

[16]

F. J. Narcowich, X. Sun and J. D. Ward, Approximation power of RBFs and their associated SBFs: a connection,, \emph{Adv. Comput. Math. 27}, 1 (2007), 107. doi: 10.1007/s10444-005-7506-1.

[17]

I. J. Schoenberg, Metric spaces and completely monotone functions,, \emph{The Annals of Mathematics 39}, 4 (1938), 811. doi: 10.2307/1968466.

[18]

I. J. Schoenberg, Positive definite functions on spheres,, \emph{Duke Math. J.}, 9 (1988), 96.

[19]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2$^{nd}$ edition, (1994). doi: 10.1007/978-1-4612-0873-0.

[20]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).

[21]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces,, Princeton Mathematical Series, (1971).

[22]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals,, 3$^{nd}$ edition, (1986).

[23]

C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations,, \emph{Discrete Continuous Dynam. Systems - B}, 4 (2004), 1065. doi: 10.3934/dcdsb.2004.4.1065.

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