论文标题
根据插值投影仪的规范的估计
On Some Estimate for the Norm of an Interpolation Projector
论文作者
论文摘要
令$ q_n = [0,1]^n $为$ {\ Mathbb r}^n $中的单位立方体，让$ c（q_n）$是连续函数的一个空间$ f：q_n \ to {\ mathbb r} $，带有norm $ \ | q_n} | f（x）|。$ by $π_1\ left（{\ mathbb r}^n \ right）$表示一组$ \ leq 1 $的多项式，即，在$ {\ mathbb r}^n $上进行了一组线性函数。插值投影仪$ p：c（q_n）\ toπ_1（{\ mathbb r}^n）$与节点$ x^{（j）} \ in q_n $定义的$ x^{（j）} \是由均等$ pf \ pf \ left（x^{（j）}} \ y pf \ weft（j）} \ right）= f \ j $ j $ weft（ $ \ ldots，$ $ n+1 $。令$ \ | p \ | _ {q_n} $是$ p $作为运算符的范围，从$ c（q_n）$到$ c（q_n）$。如果$ n+1 $是Hadamard号码，则存在一个非等级的常规单纯形，顶点为$ q_n $。我们讨论了一些方法，以获取$ || p || _ {q_n} \ leq c \ sqrt {n} $的不平等现象，以示为相应的投影仪$ p $。
Let $Q_n=[0,1]^n$ be the unit cube in ${\mathbb R}^n$ and let $C(Q_n)$ be a space of continuous functions $f:Q_n\to{\mathbb R}$ with the norm $\|f\|_{C(Q_n)}:=\max_{x\in Q_n}|f(x)|.$ By $Π_1\left({\mathbb R}^n\right)$ denote a set of polynomials of degree $\leq 1$, i.e., a set of linear functions on ${\mathbb R}^n$. The interpolation projector $P:C(Q_n)\to Π_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in Q_n$ is defined by the equalities $Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right)$, $j=1,$ $\ldots,$ $ n+1$. Let $\|P\|_{Q_n}$ be the norm of $P$ as an operator from $C(Q_n)$ to $C(Q_n)$. If $n+1$ is an Hadamard number, then there exists a nondegenerate regular simplex having the vertices at vertices of $Q_n$. We discuss some approaches to get inequalities of the form $||P||_{Q_n}\leq c\sqrt{n}$ for the norm of the corresponding projector $P$.
