An operator approach to the indefinite Stieltjes moment problem
An operator approach to the indefinite Stieltjes moment problem
复制标题
求解不定 Stieltjes 矩问题的算子方法
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
I. Kovalyov
中科院分区：
文献类型：
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作者：
V. Derkach;I. Kovalyov
如果f关于ℂℝ和核Nκ(κϵℤ+是对称的，则称f是关于ℝ和核Nωz≔fz−fω$${−ω{N}}_{omega}(Z)Coloneq Frc{f(Z)-Overline{fLeft(omega
Ight)}}{z-OVERLINE{omega}}$$在κ+上有ℂ负平方。广义Stieltjes类Nκkκk∈ℤ+$${mathbf{N}_{kappa}^kleft(kappa，kin{mathrm{mathbb{Z}})_{+}
T)$$定义为函数集fϵNκ使得z fϵNk。完全不定Stieltjes矩问题MPκKs$${MP}_{kappa}^kleft(mathbf{S}
8)$$包含如下：给定κ，kϵℤ+，且序列S=sii=0∞$$mathbf{S}={左{{S}_i
满足渐近展开的函数集f∈Nκk$$fin{mathbf{N}}_{kappa}^k$$
Fz=−s0z−⋯−s2nz2n+1+o1z2n+1z=−y∈ℝ−y↑∞$$f(Z)=-Frac{S_0}{z}-cdots-Frac{S_2n}{z^{2n+1}}+olft(Frc{1}{z^{2n+1}}
右)左(z=-阴{mathm{mathbb{R}})_{-}，yuparrow inty
夜)$$
对于所有的n足够大。在本文中，我们将解决不定Stieltjes矩问题MPκks$${MP}_{kappa}^kleft(mathbf{S}
$$)$$应用于由J0N$${mathfrak{J}}_{Left[0；N]生成的Pontryagin空间对称算子A[0；N]
夜]}$$。利用边界三元组技巧，利用广义Stieltjes多项式计算了算子A[0；N]的u-预解矩阵。MPκKs$${MP}_{kappa}^kLeft(Mathbf{S})问题的若干判据
Ight)$$是可解的和不确定的。文中还给出了广义Stieltjes分式的Padé逼近的显式公式。
AbstractA function f meromorphic on ℂℝ is said to be in the generalized Nevanlinna class Nκ (κ ϵ ℤ+), if f is symmetric with respect to ℝ and the kernel Nωz≔fz−fω¯z−ω¯$$ {mathbf{N}}_{omega }(z)coloneq frac{f(z)-overline{fleft(omega
ight)}}{z-overline{omega}} $$ has κ negative squares on ℂ+. The generalized Stieltjes class Nκkκk∈ℤ+$$ {mathbf{N}}_{kappa}^kleft(kappa, kin {mathrm{mathbb{Z}}}_{+}
ight) $$ is defined as the set of functions f ϵ Nκ such that z f ϵ Nk. The full indefinite Stieltjes moment problem MPκks$$ {MP}_{kappa}^kleft(mathbf{s}
ight) $$ consists in the following: Given κ, k ϵ ℤ+, and a sequence s=sii=0∞$$ mathbf{s}={left{{s}_i
ight}}_{i=0}^{infty } $$ of real numbers, to describe the set of functions f∈Nκk$$ fin {mathbf{N}}_{kappa}^k $$, which satisfy the asymptotic expansion
fz=−s0z−⋯−s2nz2n+1+o1z2n+1z=−y∈ℝ−y↑∞$$ f(z)=-frac{s_0}{z}-cdots -frac{s_2n}{z^{2n+1}}+oleft(frac{1}{z^{2n+1}}
ight)kern1em left(z=-yin {mathrm{mathbb{R}}}_{-},yuparrow infty
ight) $$
for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem MPκks$$ {MP}_{kappa}^kleft(mathbf{s}
ight) $$ within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A[0;N] generated by J0N$$ {mathfrak{J}}_{left[0;N
ight]} $$. The u-resolvent matrices of the operator A[0;N] are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple’s technique. Some criteria for the problem MPκks$$ {MP}_{kappa}^kleft(mathbf{s}
ight) $$ to be solvable and indeterminate are found. Explicit formulae for Padé approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.