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

Abstract:

      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 $1 \leq p \leq \infty$.