Weighted Polynomial Inequalities
Polynomial inequalities have been playing crucial roles in approximation theory and related fields. Several such inequalities on the unit sphere will be established in this chapter. Since some of them will be needed in weighted approximation theory and harmonic analysis in later chapters, we prove them in the weighted L p norm. We will work in the context of doubling weights, defined and discussed in the first section. A fundamental tool in our approach to polynomial inequalities is a maximal function for spherical polynomials, introduced and studied in the second section, which can be controlled pointwise by the Hardy–Littlewood maximal function and which possess several other useful properties. In the third section, we establish the Marcinkiewics–Zygmund inequality, which, in its most useful form, states that the norm of a polynomial with respect to a finite discrete measure, in fact a sum with well-separated points, is bounded by its L p norm on the sphere. This inequality will play an important role in the next chapter. In the fourth section, we establish the Bernstein and Nikolskii inequalities.
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