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Continuity of weighted operators, Muckenhoupt \(A_p\) weights, and Steklov problem for orthogonal polynomials. (English) Zbl 1487.42059

Summary: We consider weighted operators acting on \(L^p(\mathbb{R}^d)\) and show that they depend continuously on the weight \(w\in A_p(\mathbb{R}^d)\) in the operator topology. Then, we use this result to estimate \(L^p_w(\mathbb{T})\) norm of polynomials orthogonal on the unit circle when the weight \(w\) belongs to Muckenhoupt class \(A_2(\mathbb{T})\) and \(p>2\). The asymptotics of the polynomial entropy is obtained as an application.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42B30 \(H^p\)-spaces
47B38 Linear operators on function spaces (general)
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