Correction/Addendum to "The Extremal Function for the Complex Ball for Generalized Notions of Degree and Multivariate Polynomial Approximation"

Solution to the Khavinson problem near the boundary of the unit ball

This paper deals with an extremal problem for harmonic functions in the unit ball of $\mathbf{R}^n$. We are concerned with the pointwise sharp estimates for the gradient of real--valued bounded harmonic functions. Our main result may be formulated as follows. The sharp constants in the estimates for the absolute value of the radial derivative and the modulus of the gradient of a bounded harmonic function coincide near the boundary of the unit ball. This result partially confirms a conjecture posed by D. Khavinson.

单位球上的逼近、插值与提升

Approximation, interpolation, and lifting on the unit ball

本文通过内函数和极值函数解决了单位球上的Nevanlinna-Pick插值问题，并推广了交换子提升定理，解决了多个著名问题。

A proof of Khavinson's conjecture in $\mathbf{R}^4$

The paper deals with an extremal problem for bounded harmonic functions in the unit ball of $\mathbf{R}^4$. We solve the generalized Khavinson problem in $\mathbf{R}^4$. This precise problem was formulated by G. Kresin and V. Maz'ya for harmonic functions in the unit ball and in the half--space of $\mathbf{R}^n$. We find the optimal pointwise estimates for the norm of the gradient of bounded real--valued harmonic functions.

Extremal functions in de Branges and Euclidean spaces II

This paper presents the Gaussian subordination framework to generate optimal one-sided approximations to multidimensional real-valued functions by functions of prescribed exponential type. Such extremal problems date back to the works of Beurling and Selberg and provide a variety of applications in analysis and analytic number theory. Here we majorize and minorize (on $\mathbb{R}^N$) the Gaussian ${\bf x} \mapsto e^{-πλ|{\bf x}|^2}$, where $λ>0$ is a free parameter, by functions with distributional Fourier transforms supported on Euclidean balls, optimizing weighted $L^1$-errors. By integratin…

涉及Garsia范数的BMO和VMO中的极值问题

Extremal problems in BMO and VMO involving the Garsia norm

本文研究涉及Garsia范数的BMO和VMO空间极值问题，证明单位范数范数达到BMO函数是BMO单位球极值点，VMO单位球极值点为单位范数VMO函数，并给出内函数的几何刻画。

通过Herglotz测度与Koenigs线性化刻画极值双曲速率

Characterizations of extremal hyperbolic rates via Herglotz measures and Koenigs linearization

本文利用Herglotz测度与Koenigs线性化，刻画单位圆盘双曲全纯自映射的极值双曲速率，将理论推广至多连通域、单位球及拟共形映射，明确其增长由Denjoy-Wolff点处边界线性化非退化性决定。

On Complex (non analytic) Chebyshev Polynomials in $\bbC^2$

We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in $\bbC^2$ by polynomials of lower degree with complex coefficients. We reduce the problem to a one-dimensional weighted minimization problem on an interval. In a sense, the corresponding extremal polynomials are uniform counterparts of the classical orthogonal Jacobi polynomials. They can be represented by means of special conformal mappings on the so-called comb-like domains. In these terms, the value of the minimal deviation and the representation for a polynomial of best approximation…