$L^{p}_{[0,1]}$空间M\"{u}ntz有理逼近的Jackson型估计 (英)

设$\Lambda=\{\lambda_{n}\}_{n=1}^{\infty}$为正的实数数列, 且当$n\rightarrow\infty$时, 有$\lambda_{n}\searrow 0$.本文给出了当 $\lambda_{n}\leq Mn^{-\frac{1}{2}},n=1,2, \cdots ,$(其中$M>0$为一正常数)时M\"{u}ntz系统$\{x^{\lambda_n}\}$的有理函数在$ L_{0,1]} ^{p}$空间的逼近速度,主要结论为$R_{n} (f, \Lambda )_{L^{p}}\leq C_M \omega (f, n^{-\frac{1}{2}})_{L^{p}},1 \leq p \leq \infty.$

On Jackson Estimate for M\"{u}ntz Rational Approximation in $L^{p}_{[0,1]}$ Spaces

Let $\Lambda=\{\lambda_{n}\}_{n=1}^{\infty}$ be a sequence of real numbers, and $\lambda_{n}\searrow 0$ as $n\rightarrow\infty$. Suppose that $\lambda_{n}\leq Mn^{-\frac{1}{2}}$ for $n=1,2, \cdots$, where $M>0$ is an absolute constant. The present paper considers the M\"{u}ntz rational approximation rate in $ L_{0,1]}^{p}$ spaces and gets $R_{n} (f, \Lambda )_{L^{p}}\leq C_M \omega (f, n^{-\frac{1}{2}})_{L^{p}}$ for $1 \leq p \leq \infty$.