In this survey, the notion of a balanced best multipoint local approximation is fully exposed since they were treated in the $L^p$ spaces and recent results in Orlicz spaces. The notion of balanced point, which was introduced by Chui et al. in 1984 is extensively used.

This paper is part II of "On Copositive Approximation in Spaces of Continuous Functions".  In this paper the author shows that if $Q$ is any compact subset of real numbers, and $M$ is any finite dimensional strict Chebyshev subspace of $C(Q)$, then for any admissible function $f\in C(Q)\backslash M,$ the best copositive approximation to $f$ from $M$ is unique.

Let $p(z)$ be a polynomial of degree $n$, which has no zeros in $|z|< 1$, Dewan et al. [K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities,  J. Math. Anal. Appl., 363 (2010), pp.  38-41] established $$\Big|zp'(z)+\frac{n\beta}{2}p(z)\Big| \leq \frac{n}{2}\Big\{\Big(\Big|\frac{\beta}{2}\Big|+\Big|1+\frac{\beta}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{\beta}{2}\Big|-\Big|\frac{\beta}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$ for any $|\beta|\leq 1$ and $|z|=1$. In this paper we improve the above inequality for the polynomial which has no zeros in $|z|< k, $ $ k\geq 1$, except $s$-fold zeros at the origin. Our results generalize certain well known polynomial inequalities.

Let $({\Bbb R}_+,*,\Delta)$ be the Jacobi hypergroup. We introduce analogues of the Littlewood-Paley $g$ function and the Lusin area function for the Jacobi hypergroup and consider their $(H^1, L^1)$ boundedness. Although the $g$ operator for $({\Bbb R}_+,*,\Delta)$ possesses better property than the classical $g$ operator, the Lusin area operator has an obstacle arisen from a second convolution. Hence, in order to obtain the $(H^1, L^1)$ estimate for the Lusin area operator, a slight modification in its form is required.

Using the Gegenbauer polynomials and the zonal harmonics functions we give some representation formulae of the Green function in the annulus. We apply this result to prove some uniqueness results for some nonlinear elliptic problems.

General interpolation formulae for barycentric interpolation and barycentric rational Hermite interpolation are established by introducing multiple parameters, which include many kinds of barycentric interpolation and barycentric rational Hermite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.

Let $\mathbb{H}^n$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension. In this paper, we consider the Schrödinger operator $−∆_{\mathbb{H}^n} +V$, where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian and $V$ is the nonnegative potential belonging to the reverse Hölder class $B_{q_1}$ for $q_1 ≥ Q/2$. We show that the operators $T_1 = V(−∆_{\mathbb{H}^n} +V)^{−1}$ and $T_2 = V^{1/2}(−∆_{\mathbb{H}^n} +V)^{−1/2}$ are both bounded from $H^1_L(\mathbb{H}^n)$ into $L^1(\mathbb{H}^n)$. Our results are also valid on the stratified Lie group.

Suppose $T^{k,1}$ and $T^{k,2}$ are singular integrals with variable kernels and mixed homogeneity or $\pm I$ (the identity operator). Denote the Toeplitz type operator by\begin{align*}T^b=\sum_{k=1}^QT^{k,1}M^bT^{k,2}, \end{align*} where $M^bf=bf.$ In this paper, the boundedness of $T^b$ on weighted Morrey space are obtained when $b$ belongs to the weighted Lipschitz function space and weighted BMO function space, respectively.