A Characterization of Generalized Zeros of Negative Type of Functions of the Class Nκ
A Characterization of Generalized Zeros of Negative Type of Functions of the Class Nκ
复制标题
Nκ类负型函数的广义零点的刻画
DOI：
--
复制
发表时间：
1986
期刊：
影响因子：
--
通讯作者：
H. Langer
中科院分区：
文献类型：
--
作者：
H. Langer
Recall ([1], [2], [3]) that Nκ denotes the set of all complex valued functions Q which are meromorphic in the open upper half plane C + and such that the kernel NQ:
$${N_Q}left( {z,zeta }
ight):left( {Qleft( z
ight) - overline {Qleft( zeta
ight)} }
ight)/left( {z - overline zeta }
ight)$$
(1.1)
for z,ζ e D Q has κ negative squares (here D Q (⊂C +) denotes the domain of holomorphy of Q). This means that for arbitrary n e Z and z1,z2,...,zn e D Q the matrix (NQ(zi,zj)) 1 n has at most κ negative eigenvalues and for at least one choice of n, z1,...,zn it has exactly κ negative eigenvalues. The class No coincides with the Nevanlinna class of all functions which are holomorphic in C + and map C + into C + UR. The following two examples of functions of the class N1 were considered in [2], [4], respectively:
$$wleft( z
ight):alpha - z + intlimits_{ - infty }^infty {left( {{{left( {t - z}
ight)}^{ - 1}} - t{{left( {1 + {t^2}}
ight)}^{ - 1}}}
ight)} d{sigma _O}left( t
ight),vleft( z
ight): = alpha + left( {1/z}
ight) + intlimits_{ - 8}^infty {left( {{{left( {t - z}
ight)}^{ - 1}} - t{{left( {1 + {t^2}}
ight)}^{ - 1}}}
ight)} d{sigma _1}left( t
ight),$$
(1.2)
where α e R and σo, σl are nondecreasing functions on R such that
$${intlimits_{ - infty }^infty {left( {1 + {t^2}}
ight)} ^{ - 1}}d{sigma _j}left( t
ight) < infty ,j = 0,1,{sigma _1}left( {0 + }
ight) = {sigma _1}left( {0 - }
ight).$$
Recall ([1], [2], [3]) that Nκ denotes the set of all complex valued functions Q which are meromorphic in the open upper half plane C + and such that the kernel NQ:
$${N_Q}left( {z,zeta }
ight):left( {Qleft( z
ight) - overline {Qleft( zeta
ight)} }
ight)/left( {z - overline zeta }
ight)$$
(1.1)
for z,ζ e D Q has κ negative squares (here D Q (⊂C +) denotes the domain of holomorphy of Q). This means that for arbitrary n e Z and z1,z2,...,zn e D Q the matrix (NQ(zi,zj)) 1 n has at most κ negative eigenvalues and for at least one choice of n, z1,...,zn it has exactly κ negative eigenvalues. The class No coincides with the Nevanlinna class of all functions which are holomorphic in C + and map C + into C + UR. The following two examples of functions of the class N1 were considered in [2], [4], respectively:
$$wleft( z
ight):alpha - z + intlimits_{ - infty }^infty {left( {{{left( {t - z}
ight)}^{ - 1}} - t{{left( {1 + {t^2}}
ight)}^{ - 1}}}
ight)} d{sigma _O}left( t
ight),vleft( z
ight): = alpha + left( {1/z}
ight) + intlimits_{ - 8}^infty {left( {{{left( {t - z}
ight)}^{ - 1}} - t{{left( {1 + {t^2}}
ight)}^{ - 1}}}
ight)} d{sigma _1}left( t
ight),$$
(1.2)
where α e R and σo, σl are nondecreasing functions on R such that
$${intlimits_{ - infty }^infty {left( {1 + {t^2}}
ight)} ^{ - 1}}d{sigma _j}left( t
ight) < infty ,j = 0,1,{sigma _1}left( {0 + }
ight) = {sigma _1}left( {0 - }
ight).$$