多個複變數函數論Ⅱ 超球雙曲空間中的一完整正交函數系

<正> §1.引言 在上一篇文章裹,我曾經具體地算出矩陣的雙曲空間中的完整正交函数系,在該文中引用了方陣羣的表示法的理論.在這一篇文章裹,我們將定出超球雙曲空間中的完整正交系.所用的方法和上篇稍有不同,我們除掉用一些正交羣的表示羣以外,還用了不變量論中的結果及若干與球調和(spherical harmonic)

ON THE THEORY OF FUNCTIONS OF SEVERAL COMPLEX VARIABLES II A COMPLEX ORTHO-NORMAL SYSTEM IN THE HYPERBOLIC SPACE OF LIE-HYPERSPHERE

Let n≥2 and z=(z_1,...,z_n) be an n-dimensional complex vector, denotes the space formed by vectors satisfying (1) and (2), where z’denotes the transposed column vector obtained from z. By means of spherical harmonics and the invariants of orthogonal group and its related techniques, we obtained an orthogonal system of. Moreover, the characteristic manifold of is defined by |zz′| = 1, zz′= 1.Cauchy formula has been obtained, which asserts that the function is uniquely determined in by its values on. Consequently, we proved that the maximal of the modulus of a function analytic in and on is taken on. From it we deduce several theorems which are refinements of Hadamard's three sphere theorem.