Abstract
The fundamental theorem of harmonic analysis on compact groups is just the Peter-Weyl theorem. It claims that every continuous function on a compact group can be approximated as closely as desired by linear combinations of irreducible representations of the group. Professor Hua modified this theorem in Ref. 1. He defined Fourier series for continuous functions on the unitary group U(n) and proved that any continuous function on the unitary group can be obtained through Fourier series via Abel summability.
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References
L. K. Hua, A convergent theorem of the space of continuous functions on compact groups, Sci. Record (N.S) No. 9 (1958).
L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domain, Science Press, Beijing (1958) (in Chinese); American Mathematical Society, Providence, RI (1968).
L. K. Hua and K. H. Look, Theory of harmonic functions in the classical domain. I, Acta Math. Sinica 8, 531–547 (1958); Sci. Sinica 9, 1031–1094 (1958) (in English).
K. H. Look, Classical Manifolds and Classical Domains, Shanghai Science and Technical, Shanghai (1963).
S. Kung, Fourier analysis on unitary groups. I, Acta Math. Sinica 10, 239–261 (1960); Chinese Math. Acta 3, 249–272 (1962).
S. Kung, Fourier analysis on unitary groups. II, Acta Math. Sinica 12, 17–32 (1962); Chinese Math. Acta 3, 19–33 (1963).
E. D. Murnaghan, The Theory of Group Representations, Johns Hopkins University Press, Baltimore (1938).
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© 1991 Plenum Press, New York
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Wu, HH. (1991). Harmonic Analysis on Rotation Groups: Abel Summability. In: Wu, HH. (eds) Contemporary Geometry. The University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-7950-8_6
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DOI: https://doi.org/10.1007/978-1-4684-7950-8_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4684-7952-2
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