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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 and 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, U.S. Government Printing Office, Washington, D.C., 1964. [2] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications, No. 71. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937. [3] W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. [4] E. Belinsky, F. Dai and Z. Ditzian, Multivariate approximating averages, J. Approx. Theory 125, 1 (2003), 85-105. 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, J. Funct. Anal. 255, 9 (2008), 2265-2285. 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, Proceedings of the American Mathematical Society 131, 9 (2003), 2733-2740. doi: 10.1090/S0002-9939-03-06730-3. [8] F. Dai and Z. Ditzian, Combinations of multivariate averages, J. Approx. Theory 131, 2 (2004), 268-283. doi: 10.1016/j.jat.2004.10.003. [9] Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar. 81, 4 (1998), 323-348. doi: 10.1023/A:1006554907440. [10] Z. Ditzian, Smoothness of a function and the growth of its Fourier transform or its Fourier coefficients, J. Approx. Theory 162, 5 (2010), 980-986. doi: 10.1016/j.jat.2009.11.003. [11] Z. Ditzian, Relating smoothness to expressions involving Fourier coefficient or to a Fourier transform, J. Approx. Theory 164, 10 (2012), 1369-1389. doi: 10.1016/j.jat.2012.05.004. [12] R. E. Edwards, Fourier Series: A Modern Introduction, Vol. II, Holt, Rinehart and Winston, INC., New York, 1967. [13] D. Gioev, Moduli of continuity and average decay of Fourier transforms: two sided estimates, in Integrable Systems and Random Matrices (Cont. Math. 458), Amer. Math. Soc., Providence, RI, (2008), 377-392. doi: 10.1090/conm/458/08948. [14] D. Gorbachev and S. Tikhonov, Moduli of smoothness and growth properties of Fourier transforms: two sided estimates, J. Approx. Theory 164, 9 (2012), 1283-1312. doi: 10.1016/j.jat.2012.05.017. [15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6nd edition, translated from the Russian sixth edition, translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, Academic Press, Inc., San Diego, CA, 2000. [16] F. J. Narcowich, X. Sun and J. D. Ward, Approximation power of RBFs and their associated SBFs: a connection, Adv. Comput. Math. 27, 1 (2007), 107-124. doi: 10.1007/s10444-005-7506-1. [17] I. J. Schoenberg, Metric spaces and completely monotone functions, The Annals of Mathematics 39, 4 (1938), 811-841. doi: 10.2307/1968466. [18] I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J., 9 (1988), 96-108. [19] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970. [21] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971. [22] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 3nd edition, Chelsea Publishing Co., New York, 1986. [23] C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems - B, 4 (2004), 1065-1089. doi: 10.3934/dcdsb.2004.4.1065.

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##### 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, U.S. Government Printing Office, Washington, D.C., 1964. [2] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications, No. 71. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937. [3] W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. [4] E. Belinsky, F. Dai and Z. Ditzian, Multivariate approximating averages, J. Approx. Theory 125, 1 (2003), 85-105. 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, J. Funct. Anal. 255, 9 (2008), 2265-2285. 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, Proceedings of the American Mathematical Society 131, 9 (2003), 2733-2740. doi: 10.1090/S0002-9939-03-06730-3. [8] F. Dai and Z. Ditzian, Combinations of multivariate averages, J. Approx. Theory 131, 2 (2004), 268-283. doi: 10.1016/j.jat.2004.10.003. [9] Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar. 81, 4 (1998), 323-348. doi: 10.1023/A:1006554907440. [10] Z. Ditzian, Smoothness of a function and the growth of its Fourier transform or its Fourier coefficients, J. Approx. Theory 162, 5 (2010), 980-986. doi: 10.1016/j.jat.2009.11.003. [11] Z. Ditzian, Relating smoothness to expressions involving Fourier coefficient or to a Fourier transform, J. Approx. Theory 164, 10 (2012), 1369-1389. doi: 10.1016/j.jat.2012.05.004. [12] R. E. Edwards, Fourier Series: A Modern Introduction, Vol. II, Holt, Rinehart and Winston, INC., New York, 1967. [13] D. Gioev, Moduli of continuity and average decay of Fourier transforms: two sided estimates, in Integrable Systems and Random Matrices (Cont. Math. 458), Amer. Math. Soc., Providence, RI, (2008), 377-392. doi: 10.1090/conm/458/08948. [14] D. Gorbachev and S. Tikhonov, Moduli of smoothness and growth properties of Fourier transforms: two sided estimates, J. Approx. Theory 164, 9 (2012), 1283-1312. doi: 10.1016/j.jat.2012.05.017. [15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6nd edition, translated from the Russian sixth edition, translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, Academic Press, Inc., San Diego, CA, 2000. [16] F. J. Narcowich, X. Sun and J. D. Ward, Approximation power of RBFs and their associated SBFs: a connection, Adv. Comput. Math. 27, 1 (2007), 107-124. doi: 10.1007/s10444-005-7506-1. [17] I. J. Schoenberg, Metric spaces and completely monotone functions, The Annals of Mathematics 39, 4 (1938), 811-841. doi: 10.2307/1968466. [18] I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J., 9 (1988), 96-108. [19] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970. [21] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971. [22] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 3nd edition, Chelsea Publishing Co., New York, 1986. [23] C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems - B, 4 (2004), 1065-1089. doi: 10.3934/dcdsb.2004.4.1065.
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