Rayleigh triangles and non-matrix interpolation of matrix beta-integrals

六边形和三角形上的离散傅里叶分析、求积与插值

Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle

利用离散傅里叶变换和二元正交多项式，研究六边形和三角形上的三角逼近问题，详细发展了正六边形上的离散傅里叶分析，并推导出三角形上的分析，包括求积公式和插值，其中三角形上的三角拉格朗日插值满足显式紧致公式，等价于斯坦纳内摆线所围平面区域上的多项式插值，插值的勒贝格常数为(\log n)^2阶，并在内摆线上建立了高斯求积。

Inequalities for $L^p$-norms that sharpen the triangle inequality and complement Hanner's Inequality

In 2006 Carbery raised a question about an improvement on the naïve norm inequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$ this is an equality, but when the supports of $f$ and $g$ are disjoint the factor $2^{p-1}$ is not needed. Carbery's question concerns a proposed interpolation between the two situations for $p>2$. The interpolation parameter measuring the overlap is $\|fg\|_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all $…