This book treats spherical harmonic expansion of real analytic functions and hyperfunctions on the sphere. Because a one-dimensional sphere is a circle, the simplest example of the theory is that of Fourier series of periodic functions. The author first introduces a system of complex neighborhoods of the sphere by means of the Lie norm. He then studies holomorphic functions and analytic functionals on the complex sphere. In the one-dimensional case, this corresponds to the study of holomorphic functions and analytic functionals on the annular set in the complex plane, relying on the Laurent series expansion. In this volume, it is shown that the same idea still works in a higher-dimensional sphere. The Fourier-Borel transformation of analytic functionals on the sphere is also examined; the eigenfunction of the Laplacian can be studied in this way.
First introduces a system of complex neighborhoods of the sphere by means of the Lie norm, then examines holomorphic and analytic functions on the complex sphere. In the one-dimensional case of a sphere<-->a circle<-->this corresponds to the study of holomorphic and analytic functions on the annular set in the complex plane, relying on the Laurent series expansion, and Morimoto shows that the same idea works on higher-dimensional spheres as well. Graduate students, research mathematicians, and mathematical physicists may find the treatment of interest. Annotation c. by Book News, Inc., Portland, Or.
