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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

Thaís Jordão 1, and  Xingping Sun 2,

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.
Keywords: $K$-funcional, Laplacians of fractional order, Fourier multiplier, Bochner-Riesz means, spherical means..
Mathematics Subject Classification: Primary: 40A05, 41A36; Secondary: 41A60, 45C05, 47A7.
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, U.S. Government Printing Office, Washington, D.C., 1964.  Google Scholar

[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.  Google Scholar

[3]

W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar. 81, 4 (1998), 323-348. doi: 10.1023/A:1006554907440.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. E. Edwards, Fourier Series: A Modern Introduction, Vol. II, Holt, Rinehart and Winston, INC., New York, 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

I. J. Schoenberg, Metric spaces and completely monotone functions, The Annals of Mathematics 39, 4 (1938), 811-841. doi: 10.2307/1968466.  Google Scholar

[18]

I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J., 9 (1988), 96-108.  Google Scholar

[19]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[20]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[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.  Google Scholar

[22]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 3nd edition, Chelsea Publishing Co., New York, 1986.  Google Scholar

[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.  Google Scholar

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, U.S. Government Printing Office, Washington, D.C., 1964.  Google Scholar

[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.  Google Scholar

[3]

W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

Z. Ditzian, Fractional derivatives and best approximation, Acta Math. Hungar. 81, 4 (1998), 323-348. doi: 10.1023/A:1006554907440.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. E. Edwards, Fourier Series: A Modern Introduction, Vol. II, Holt, Rinehart and Winston, INC., New York, 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

I. J. Schoenberg, Metric spaces and completely monotone functions, The Annals of Mathematics 39, 4 (1938), 811-841. doi: 10.2307/1968466.  Google Scholar

[18]

I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J., 9 (1988), 96-108.  Google Scholar

[19]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[20]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[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.  Google Scholar

[22]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 3nd edition, Chelsea Publishing Co., New York, 1986.  Google Scholar

[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.  Google Scholar

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