| 熊静宜,曹飞龙,杨汝月.球面Jackson多项式逼近的正逆定理[J].数学年刊A辑,2011,32(2):205~212 |
| 球面Jackson多项式逼近的正逆定理 |
| The Direct and Inverse Theorem of Approximation for Jackson Polynomials on the Sphere |
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| DOI： |
| 中文关键词: 球面 Jackson多项式 逼近 光滑模 下界 |
| 英文关键词:Sphere Jackson polynomials Approximation Modulus of smoothness Lower bound |
| 基金项目:国家自然科学基金(No.60873206)资助的项目 |
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| 中文摘要: |
| 研究了球面Jackson多项式$J_{v,s}f$的逼近阶, 建立了该多项式逼近的强型正向与逆向不等式.
利用球面光滑模较好地刻画了\!Jackson多项式的逼近性能,
证明了存在与$v$和$f$无关的常数$C_1$和$C_2$,使得对于定义在球面上任意$p$-幂勒贝格可积或连续函数$f$成立
$$
C_1\omega\Big(f,\frac{1}{v}\Big)_p \leq \|J_{v,s}f-f\|_p \leq C_2\omega\Big(f,\frac{1}{v}\Big)_p,
$$
其中$\omega(f,t)_p$是$f$的光滑模. |
| 英文摘要: |
| The degree of approximation for the Jackson polynomials $J_{v,s}f$ on the unit sphere is considered. The direct and inverse inequalities
of strong-type for approximation by these polynomials are established. The approximation behavior of these polynomials is nicely characterized
by using modulus of smoothness of sphere. Namely, it is proved that there exist constants $C_1$ and $C_2$ independent
of $v$ and $f$ such that
$$
C_1\omega\Big(f,\frac{1}{v}\Big)_p \leq \|J_{v,s}f-f\|_p \leq C_2\omega\Big(f,\frac{1}{v}\Big)_p
$$
for any $p$-power Lebesgue integrable or continuous function $f$ defined on the sphere, where $\omega(f,t)_p$ is the modulus of smoothness of $f$. |
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