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

Previous Article
Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities
CPAA Home
This Issue
Next Article

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

Abstract
Related Papers

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 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.

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.
[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.

[1]	Chenchen Wu, Wei Lv, Yujie Wang, Dachuan Xu. Approximation algorithm for spherical $ k $-means problem with penalty. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021067
[2]	Sung Ha Kang, Berta Sandberg, Andy M. Yip. A regularized k-means and multiphase scale segmentation. Inverse Problems and Imaging, 2011, 5 (2) : 407-429. doi: 10.3934/ipi.2011.5.407
[3]	Baolan Yuan, Wanjun Zhang, Yubo Yuan. A Max-Min clustering method for $k$-means algorithm of data clustering. Journal of Industrial and Management Optimization, 2012, 8 (3) : 565-575. doi: 10.3934/jimo.2012.8.565
[4]	Baoli Shi, Zhi-Feng Pang, Jing Xu. Image segmentation based on the hybrid total variation model and the K-means clustering strategy. Inverse Problems and Imaging, 2016, 10 (3) : 807-828. doi: 10.3934/ipi.2016022
[5]	Min Li, Yishui Wang, Dachuan Xu, Dongmei Zhang. The approximation algorithm based on seeding method for functional $ k $-means problem^†. Journal of Industrial and Management Optimization, 2022, 18 (1) : 411-426. doi: 10.3934/jimo.2020160
[6]	Fan Yuan, Dachuan Xu, Donglei Du, Min Li. An exact algorithm for stable instances of the $ k $-means problem with penalties in fixed-dimensional Euclidean space. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021122
[7]	Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131
[8]	Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems and Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036
[9]	Marius Tucsnak. Control of plate vibrations by means of piezoelectric actuators. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 281-293. doi: 10.3934/dcds.1996.2.281
[10]	Gilles Carbou, Stéphane Labbé, Emmanuel Trélat. Smooth control of nanowires by means of a magnetic field. Communications on Pure and Applied Analysis, 2009, 8 (3) : 871-879. doi: 10.3934/cpaa.2009.8.871
[11]	Guojun Gan, Qiujun Lan, Shiyang Sima. Scalable clustering by truncated fuzzy $c$-means. Big Data & Information Analytics, 2016, 1 (2&3) : 247-259. doi: 10.3934/bdia.2016007
[12]	Liran Rotem. Banach limit in convexity and geometric means for convex bodies. Electronic Research Announcements, 2016, 23: 41-51. doi: 10.3934/era.2016.23.005
[13]	Xiaoxi Li, Marc Quincampoix, Jérôme Renault. Limit value for optimal control with general means. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2113-2132. doi: 10.3934/dcds.2016.36.2113
[14]	Maha Daoud, El Haj Laamri. Fractional Laplacians : A short survey. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 95-116. doi: 10.3934/dcdss.2021027
[15]	Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037
[16]	Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5495-5508. doi: 10.3934/dcdsb.2020355
[17]	Giovanni Covi, Keijo Mönkkönen, Jesse Railo. Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems. Inverse Problems and Imaging, 2021, 15 (4) : 641-681. doi: 10.3934/ipi.2021009
[18]	Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems and Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018
[19]	Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767
[20]	Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Thaís Jordão Xingping Sun
on Google Scholar
Thaís Jordão Xingping Sun

[Back to Top]