Multivariate weighted bernstein-type inequality and its applications

Bernstein inequality played an important role in approximation theory and Fourier analysis. This article first introduces a general system of functions and the so-called multivariate weighted Bernstein, Nikol'skii, and Ul'yanov-type inequalities. Then, the relations among these three inequalities are discussed. Namely, it is p oved 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.