希尔伯特空间中自伴算子函数的Stechkin问题，Taikov型不等式及其应用
Stechkin's problem for functions of a self-adjoint operator in a Hilbert space, Taikov-type inequalities and their applications
摘要：
本文解决了泛函$(φ(A)x, f)$（其中$φ(A)$是自伴算子$A$的某个函数）在希尔伯特空间的元素类上的逼近问题，该空间是借助于算子$A$的另一个函数$ψ(A)$定义的.此外，我们还利用$\| ψ(A)x\|$和$\| x\|$得到了一系列估计$|(φ(A)x, f)|$的尖锐Taikov型加法不等式.我们还提出了几个应用所获得的结果。首先，我们发现尖锐的常数在H ${\rm{\ddot{o}}}$ rmander定理中使用的类型的不等式比较的运营商的情况下，当运营商是在希尔伯特空间中的作用，是一个自伴算子的功能。作为另一个应用，我们获得Taikov型不等式的功能的运营商$\frac1i \frac {d}{dt}$在空间$L_2(\RR)$和$L_2(\TT)$，以及积分的谱措施，定义的帮助下，经典的正交多项式。
英文摘要：
In this paper we solve the problem of approximating functionals $(φ(A)x, f)$ (where $φ(A)$ is some function of self-adjoint operator $A$) on the class of elements of a Hilbert space that is defined with the help of another function $ψ(A)$ of the operator $A$. In addition, we obtain a series of sharp Taikov-type additive inequalities that estimate $|(φ(A)x, f)|$ with the help of $\| ψ(A)x\|$ and $\| x\|$. We also present several applications of the obtained results. First, we find sharp constants in inequalities of the type used in H${\rm{\ddot{o}}}$rmander theorem on comparison of operators in the case when operators are acting in a Hilbert space and are functions of a self-adjoint operator. As another application we obtain Taikov-type inequalities for functions of the operator $\frac1i \frac {d}{dt}$ in the spaces $L_2(\RR)$ and $L_2(\TT)$, as well as for integrals with respect to spectral measures, defined with the help of classical orthogonal polynomials.
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