Marcinkiewicz–Zygmund Inequalities and Polynomial Approximation from Scattered Data on SO(3)

REFERENCES

• Å . Björck ( 1996 ). Numerical Methods for Least Squares Problems . SIAM , Philadelphia .
   Google Scholar
• K.G. van den Boogaart , R. Hielscher , J. Prestin , and H. Schaeben ( 2007 ). Kernel-based methods for inversion of the Radon transform on SO(3) and their applications to texture analysis . J. Comp. Appl. Math. 199 ( 1 ): 122 – 140 .
   Web of Science ®Google Scholar
• A. Böttcher , S. Kunis , and D. Potts ( 2008 ). Probabilistic spherical Marcinkiewicz–Zygmund inequalities . J. Approx. Theory . In press .
   Google Scholar
• H.J. Bunge ( 1982 ). Texture Analysis in Material Science . Butterworths , London .
   Google Scholar
• J.E. Castrillon-Candas , V. Siddavanahalli , and C. Bajaj ( 2005 ). Nonequispaced Fourier Transforms for Protein–Protein Docking . ICES Report 05–44 , University of Texas at Austin .
   Google Scholar
• H. Dym and H.P. McKean (1972). Fourier Series and Integrals . Academic Press , New York.
   Google Scholar
• R.E. Edwards ( 1967 ). Fourier Series. A Modern Introduction . Vol. II . Holt, Rinehart and Winston, Inc. , New York .
   Google Scholar
• W. Erb and F. Filbir ( 2008 ). Approximation by positive definite functions on compact groups . Numer. Funct. Anal. Optim . In press .
   Google Scholar
• H.G. Feichtinger , K. Gröchenig and T. Strohmer ( 1995 ). Efficient numerical methods in non-uniform sampling theory . Numer. Math. 69 ( 4 ): 423 – 440 .
   Google Scholar
• F. Filbir and D. Schmid ( 2008 ). Stability results for approximation by positive definite functions on SO(3) . J. Approx. Theory . In press .
   Google Scholar
• F. Filbir and W. Themistoclakis ( 2008 ). Polynomial approximation on the sphere using scattered data . Math. Nachr. 281 ( 5 ): 650 – 668 .
   Web of Science ®Google Scholar
• I.M. Gelfand , R.A. Minlos , and Z.Ya. Shapiro ( 1963 ). Representations of the Rotation and Lorentz Groups and their Applications . Pergamon Press , Oxford .
   Google Scholar
• M. Gräf and S. Kunis ( 2008 ). Stability results for scattered data interpolation on the rotation group . Electron. Trans. Numer. Anal . In press .
   Google Scholar
• K. Gröchenig ( 1992 ). Reconstruction algorithms in irregular sampling . Math. Comput. 59 : 181 – 194 .
   Web of Science ®Google Scholar
• T. Gutzmer ( 1996 ). Interpolation by positive definite functions on locally compact groups with application to SO(3) . Res. Math. 29 : 69 – 77 .
   Google Scholar
• J. Keiner , S. Kunis , and D. Potts ( 2007 ). Efficient reconstruction of functions on the sphere from scattered data . J. Fourier Anal. Appl. 13 : 435 – 458 .
   Web of Science ®Google Scholar
• R. Lasser ( 1996 ). Introduction to Fourier Series . Marcel Dekker , New York .
   Google Scholar
• D.S. Lubinsky ( 1998 ). Marcinkiewicz–Zygmund inequalities: methods and results . In: Recent Progress in Inequalities ( et al. . eds.), Kluwer Academic Publishers , Dordrecht , pp. 213 – 240 .
   Google Scholar
• J. Marcinkiewicz and A. Zygmund ( 1937 ). Mean values of trigonometric polynomials . Fund. Math. 28 : 131 – 167 .
   Google Scholar
• G. Mastroianni and M. G. Russo ( 1999 ). Weighted Marcinkiewicz–Zygmund inequalities and boundedness of the Lagrange operator . In: Mathematical Analysis and Applications ( T.M. Rassias , ed.), Hadronic Press , Palm Harbor , FL , pp. 149 – 182 .
   Google Scholar
• H.N. Mhaskar ( 2004 ). Polynomial operators and local smoothness classes on the unit interval . J. Approx. Theory 131 ( 2 ): 243 – 267 .
   Google Scholar
• H.N. Mhaskar , F.J. Narcowich , N. Sivakumar , and J. D. Ward . ( 2002 ). Approximation with interpolatory constraints . Proc. Am. Math. Soc. 130 ( 5 ): 1355 – 1364 .
   Google Scholar
• H.N. Mhaskar , F.J. Narcowich , and J.D. Ward ( 2001 ). Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature . Math. Comput. 70 ( 235 ): 1113 – 1130 .
   Google Scholar
• H.N. Mhaskar and J. Prestin ( 1998 ). On Marcinkiewicz–Zygmund-type inequalities . In: Approximation Theory : In Memory of A.K. Varma ( et al. ., eds.), Marcel Dekker , New York , pp. 389 – 404 .
   Google Scholar
• D. Potts , J. Prestin , and A. Vollrath ( 2007 ). A Fast Fourier Algorithm on the Rotation Group . Preprint A-07-06 , Universität zu Lübeck .
   Google Scholar
• H. Schaeben , R. Hielscher , J.J. Fundenberger , D. Potts , and J. Prestin ( 2007 ). Orientation density function-controlled pole probability density function measurements: automated adaptive control of texture goniometers . J. Appl. Cryst. 40 : 570 – 579 .
   Google Scholar
• N. Vilenkin ( 1968 ). Special Functions and the Theory of Group Representations . Translations of mathematical monographs , , 22nd edition . American Mathematical Society , Providence , RI .
   Google Scholar
• A. Yershova and S.M. LaValle ( 2004 ). Deterministic sampling methods for spheres and SO(3) . In: Proceedings. ICRA 04 . IEEE International Conference on Robotics and Automation , New Orleans , LA , volume 4 , pp. 3974 – 3980 .
   Google Scholar