Abstract
We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter’s Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini’s type pointwise convergence theorem are proved.
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References
Zygmund A. Trigonometric Series, Vol 2. 2nd ed, Cambridge: Cambridge Univ Press, 1959
Carleson L. On convergence and growth of partial sums of Fourier series. Acta Math, 1966, 116: 135–157
Hunt A. On the convergence of Fourier series, orthogonal expansions and their continuous analogues. In: Proc Conf Edwardsville I11 1967. Carbondale: Southern Illinois Univ Press, Carbondale, 1968, 235–255
Roetman E L. Pointwise convergence for expansios in surface harmonics of arbitrary dimension. J Reine Angew Math, 1976, 282: 1–10
Wang K Y, Li L Q. Harmonic analysis and approximation on the unit sphere. Beijing/New York: Science Press, 2000
Dirichlet P G L. Sur les séries dont le terme général dépend de deux angle, et qui servent á exprimer des fonctions arbitraires entre des limites données. J Reine Angew Math, 1873, 17: 35–56
Meaney C. Divergence Jacobi polynomial series. Proceedings of the American Mathematical Society, 1983, 87(3): 459–462
Qian T. Singular integrals on star-shaped Lipschitz surfaces in the quaternionic space. Math Ann, 1998, 310(4): 601–630
Qian T. Fourier analysis on star-shaped Lipschitz surfaces. J of Func Anal, 2001, 183: 370–412
Liu S, Qian T. Pointwise convergence of Fourier series on the unit sphere of R 4 with Quaternionic setting. Trends in Mathematics: Advance in Analysis and Geometry. Basel: Birkhäuser, 2004, 131–147
Qian T. Generalization of Fueter’s result to R n+1. Rend Mat Acc Lincei, 1997, 8(9): 111–117
Qian T, Sommen F. Deriving harmonic functions in higher dimensional spaces. Z Anal Anwendungen, 2003, 22(2): 275–288
Brackx F, Delanghe R, Sommen F. Clifford analysis. Research Notes in Mathematics, Vol 76. Boston London Melbourne: Pitman Advanced Publishing Company, 1982, Boston, London, Melbourne
Delanghe R, Sommen F, Soucek V. Clifford algebra and spinor valued functions. A Function Theory for Dirac Operator, Dordrecht: Kluwer, 1992
Deavours C A. The quaternion calculus. Amer Math, 1973, 80: 995–1008
Sudbery A. Quaternionic analysis. Math Proc Camb Phil Soc, 1979, 85: 199–225
Sce M. Osservazioni sulle serie di potenze nei moduli quadratici. Atti Acc, Lincei Rend, 1957, 23: 220–225
Kou K I, Qian T, Sommen F. Generalizations of Futer’s Theorem. Methods Appl Anal, 2002, 9(2): 273–289
Rinehart R F. Elements of theory of intrinsic functions on algebras. Duke Math J, 1965, 32: 1–19
Turri T. A ptoposito degli automorfismi del corpo complesso, Rendiconti del Seminario della Facoltádi Scienze della Universitádi Cagliari, 1947, 17: 88–94
Fueter R. Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen. Comm Math Helv, 1935, 7: 307–330.
Koschmieder L. Unmittelbarer beweis der Konvergenz einiger riehen, die von mehvern veränderlichen abhängen. Manatsh Math Phys, 1934, 41: 58–63
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Dedicated to Professor Sheng GONG on the occasion of his 75th birthday
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Fei, M., Qian, T. Clifford algebra approach to pointwise convergence of Fourier series on spheres. SCI CHINA SER A 49, 1553–1575 (2006). https://doi.org/10.1007/s11425-006-2053-x
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DOI: https://doi.org/10.1007/s11425-006-2053-x