Continuity of Weighted Operators, Muckenhoupt Ap Weights, and Steklov Problem for Orthogonal Polynomials

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.

Weighted Inequality

In this paper, we define Ap weights, briefly discuss the theory of weighted inequalities and its application and importance in various fields. We then prove that for an Ap weight function w and for some , the function, min(w, k) is an Ap weight function. Finally we establish the weighted inequality for min(w, k).DOI: http://dx.doi.org/10.3126/jie.v10i1.10887Journal of the Institute of Engineering, Vol. 10, No. 1, 2014 pp. 121–124

Approximations in several variables with freud-type and Ap-weights

We present direct (Jackson-type) and converse (Bernstein-Stechkin-type) theorems for polynomial approximations with Freud-type weights and trigonometric approximations with Ap-weights, in the case of several variables.