摘要
Bernstein inequality played an important role in approximation theory and Fourier analysis. This article first introduces a general system of functions and the socalled multivariate weighted Bernstein, Nikol＇skii, and Ul＇yanov-type inequalities. Then, the relations among these three inequalities are discussed. Namely, it is proved that a family of functions equipped with Bernstein-type inequality satisfies Nikol＇skii-type and Ul＇yanov-type inequality. Finally, as applications, some classical inequalities are deduced from the obtained results.
Bernstein inequality played an important role in approximation theory and Fourier analysis. This article first introduces a general system of functions and the socalled multivariate weighted Bernstein, Nikol＇skii, and Ul＇yanov-type inequalities. Then, the relations among these three inequalities are discussed. Namely, it is proved that a family of functions equipped with Bernstein-type inequality satisfies Nikol＇skii-type and Ul＇yanov-type inequality. Finally, as applications, some classical inequalities are deduced from the obtained results.
基金
supported by the National Natural Science Foundation of China (90818020,60873206)
the Foundation of Innovation Team of Science and Technology of Zhejiang Province of China (2009R50024)
