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ON CONVERGENCE PROPERTIES OF 3D SPHEROIDAL MONOGENICS
Abstract
Morais has recently introduced certain complete orthogonal sets of monogenic polynomials over 3D prolate spheroids with remarkable properties. The underlying functions take on either values in the reduced and full quaternions (identified, respectively, with ℝ3 and ℝ4), and are generally assumed to be nullsolutions of the well known Riesz and Moisil–Théodoresco systems in ℝ3. In continuation of these studies, we recall some fundamental properties of the polynomials, and prove some recursive formulae between them. As a consequence, we obtain a two-term type recurrence relation satisfied by those basis polynomials. These results are then employed to investigate a rather wide class of approximation properties for monogenic functions over 3D prolate spheroids in terms of spheroidal monogenics.
Keywords:
AMSC: 30G35
References
-
A. Bitsadze , Boundary Value Problems for Second-Order Elliptic Equations ( North-Holland , Amsterdam , 1968 ) . Google Scholar - Math. Methods Appl. Sci. 32(2), 223 (2009). Crossref, ISI, Google Scholar
- J. Comput. Phys. 199(2), 688 (2004). Crossref, ISI, Google Scholar
- I. Cação, Constructive approximation by monogenic polynomials, Ph.D. diss. (Universidade de Aveiro, 2004) . Google Scholar
- Complex Variables Elliptic Equations 51(11), 847 (2006). Crossref, Google Scholar
- Numer. Algor. 55(3), 191 (2010). Crossref, ISI, Google Scholar
- SIAM J. Numer. Anal. 43(5), 1912 (2005). Crossref, ISI, Google Scholar
- Adv. Appl. Clifford Alg. 21(3), 521 (2011). Crossref, ISI, Google Scholar
- Complex Variables Elliptic Equations 52(11), 1047 (2007). Crossref, Google Scholar
- Computational Methods Function Theory 9(1), 199 (2009). Crossref, Google Scholar
- A. Dzhuraev, Singular Integral Equation Method (Moscow,: Nauka, 1987) (in Russian); Engl. transl. Longman Sci. Tech. (Harlow and Wiley, N.Y., 1992) . Google Scholar
-
C. Flammer , Spheroidal Wave Functions ( Stanford University Press , Stanford, CA , 1957 ) . Google Scholar - Comment. Math. Helv. 4 , 9 ( 1932 ) . Crossref, Google Scholar
- R. Fueter, Functions of a hyper complex variable, Lecture Notes Written and Supplemented by E. Bareiss (Math. Inst. Univ. Zürich, Fall Semester, 1949) . Google Scholar
- Pacific J. Math. 3(3), 585 (1953). Crossref, Google Scholar
-
K. Gürlebeck and W. Sprössig , Quaternionic Analysis and Elliptic Boundary Value Problems ( Akademie-Verlag , Berlin , 1989 ) . Google Scholar -
K. Gürlebeck and W. Sprössig , Quaternionic Calculus for Engineers and Physicists ( John Wiley and Sons , Chichester , 1997 ) . Google Scholar - Mathematics and Computers in Simulation 44(4), 401 (1997). Crossref, ISI, Google Scholar
-
K. Gürlebeck , K. Habetha and W. Sprößig , Holomorphic Functions in the Plane and n-dimensional Space ( Birkhäuser-Verlag , Basel, Boston, Berlin , 2008 ) . Google Scholar K. Gürlebeck and J. Morais , Local properties of monogenic mappings, AIP Conference Proceedings, Numerical Analysis and Applied Mathematics1168 (2009) pp. 797–800. Google Scholar- , Geometric Algebra Computing: In Engineering and Computer Science, eds.
E. Bayro-Corrochano and G. Scheuermann (Springer, 2010) pp. 327–342. Crossref, Google Scholar -
E. Hobson , The Theory of Spherical and Ellipsoidal Harmonics ( Cambridge , 1931 ) . Google Scholar - Mathematical Methods Appl. Sci. ( 2012 ) , DOI: 10.1002/mma.2657 . Google Scholar
-
V. Kravchenko and M. Shapiro , Integral Representations for Spatial Models of Mathematical Physics ,Research Notes in Mathematics ( Pitman Advanced Publishing Program , London , 1996 ) . Google Scholar -
V. Kravchenko , Applied Quaternionic Analysis ,Research and Exposition in Mathematics 28 ( Heldermann Verlag , Lemgo , 2003 ) . Google Scholar R. Lávicka , Generalized Appell property for the Riesz system in dimension 3, AIP Conference Proceedings1389 (2011) pp. 291–294. Google Scholar-
N. Lebedev , Special Functions and Their Applications ( Dover , New York , 1972 ) . Google Scholar - , Quaternionic Analysis in ℝ3 Versus its Hyperbolic Modification ,
NATO Science Series II. Mathematics, Physics and Chemistry 25 , eds.F. Brackx , J. S. R. Chisholm and V. Soucek ( Kluwer Academic Publishers , Dordrecht, Boston, London , 2001 ) . Google Scholar - Math. Methods Appl. Sci. 33(4), 423 (2010). ISI, Google Scholar
- Appl. Math. Lett. 23(10), 1174 (2010). Crossref, ISI, Google Scholar
- Matematica (Cluj) 5 , 142 ( 1931 ) . Google Scholar
- J. Morais, Approximation by homogeneous polynomial solutions of the Riesz system in ℝ3, Ph.D. diss. (Bauhaus-Universität, Weimar, 2009) . Google Scholar
- Complex Variables Elliptic Equations 57(5), 505 (2012). Crossref, ISI, Google Scholar
- Math. Methods Appl. Sci. 34(12), 1472 (2011). Crossref, ISI, Google Scholar
- Math. Methods Appl. Sci. 34(14), 1685 (2011). Crossref, ISI, Google Scholar
- J. Numerical Anal. Indust. Appl. Math. 6 , 105 ( 2011 ) . ISI, Google Scholar
- Math. Computer Modelling 57(4), 425 (2013). Crossref, ISI, Google Scholar
- Math. Methods Appl. Sci. ( 2012 ) , DOI: 10.1002/mma.2625 . Google Scholar
- Comput. Math. Appl. ( 2012 ) , DOI: 10.1016/j.camwa.2012.10.011 . Google Scholar
- J. Opt. Soc. Am. A 22(3), 460 (2005). Crossref, Google Scholar
-
M. Riesz , Clifford Numbers and Spinors ,Institute for Physical Science 54 ( Kluwer Academic Publishers , Dorrecht , 1993 ) . Crossref, Google Scholar - Complex Variables 27 , 17 ( 1995 ) . Crossref, Google Scholar
- Complex Variables 27 , 67 ( 1995 ) . Crossref, Google Scholar
- Bell System Tech. J. 40 , 43 ( 1961 ) . Crossref, ISI, Google Scholar
- Bell System Tech. J. 43 , 3009 ( 1964 ) . Crossref, ISI, Google Scholar
- Acta Math. 103 , 25 ( 1960 ) . Crossref, ISI, Google Scholar
- Amer. J. Math. 90 , 163 ( 1968 ) . Crossref, ISI, Google Scholar
- Math. Proc. Cambridge Phil. Soc. 85 , 199 ( 1979 ) . Crossref, ISI, Google Scholar
- Appl. Comput. Harmon. Anal. 19(3), 432 (2005). Crossref, ISI, Google Scholar
- J. Fourier Anal. Appl. 11(1), 73 (2005). Crossref, ISI, Google Scholar
- J. Fourier Anal. Appl. 10(1), 1 (2004). Crossref, ISI, Google Scholar
- Proc. Amer. Math. Soc. 135(7), 2193 (2007). Crossref, ISI, Google Scholar
Published: 31 May 2013
