Best approximation in the mean by analytic and harmonic functions

The best bounds of harmonic sequence

For any natural number $n\in\mathbb{N}$, $ \frac{1}{2n+\frac1{1-γ}-2}\le \sum_{i=1}^n\frac1i-\ln n-γ<\frac{1}{2n+\frac13}, $ where $γ=0.57721566490153286...m$ denotes Euler's constant. The constants $\frac{1}{1-γ}-2$ and $\frac13$ are the best possible. As by-products, two double inequalities of the digamma and trigamma functions are established.

调和Bergman空间上解析与共解析Toeplitz算子的交换性

Commuting analytic and co-analytic Toeplitz operators on the harmonic Bergman space

本文通过构造二次型，证明了调和Bergman空间上解析与共解析Toeplitz算子交换时至少一个符号为常数，解决了1999年的开放问题。

A note on mean-value properties of harmonic functions on the hypercube

For functions defined on the $n$-dimensional hypercube $I_n (r) = \{\bm{x} \in \mathbb{R}^n ~\vert~ \vert x_i \vert \le r,~ i = 1, 2, \ldots , n\}$ and harmonic therein, we establish certain analogues of Gauss surface and volume mean-value formulas for harmonic functions on the ball in $\mathbb{R}^n$ and their extensions for polyharmonic functions. In particular, our results contribute to the best one-sided $L^1$-approximation by harmonic functions on $I_n (r)$.

加权 $H$-调和 Bergman 空间的解析延拓

Analytic continuation of weighted $H$-harmonic Bergman spaces

本文部分解决了 Blaschke 等人提出的关于加权 $H$-调和 Bergman 空间解析延拓的问题，识别了离散 Wallach 集并揭示了结构依赖于维数奇偶性。

Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid: an analytic continuation viewpoint

In a previous paper $[$B,V-1$]$, an algebra of holomorphic ``perikernels'' on a complexified hyperboloid $ X^{(c)}_{d-1} $ (in $\Bbb C^d)$ has been introduced; each perikernel $ {\cal K} $ can be seen as the analytic continuation of a kernel $ {\bf K} $ on the unit sphere $ \Bbb S^{d-1} $ in an appropriate ``cut-domain'' , while the jump of $ {\cal K} $ across the corresponding ``cut'' defines a Volterra kernel $ K $ (in the sense of J. Faraut $\lbrack$Fa-1$\rbrack$) on the one-sheeted hyperboloid $ X_{d-1} $ (in $ \Bbb R^d). $ \par In the present paper, we obtain results of harmonic analysis …

On duality of spaces of harmonic vector fields

A differential form defined on a Riemannian manifold is said to harmonic if it is closed and co-closed. Harmonic differential forms are a natural multi-dimensional extension of the concept of analytic function of complex variable. In this paper we characterize continuous linear functionals acting of the space of germs of harmonic differential forms on a compact set. This result provides a multi-dimensional analog of a theorem by G. Köthe on the dual of the space of germs of analytic functions of complex variables on a compact.