On generalized Christoffel functions

Abstract

The generalized Christoffel function λ p,q,n (dμ;x) (0<p<∞, 0≦q<∞) with respect to a measure dμ on R is defined by $$\lambda_{p,q,n}(d\mu;x)=\inf_{Q\in\mathbf{P}_{n-1},\ Q(x)=1}\int_{\mathbf{R}} \big|Q(t)\big|^p {|t-x|}^q\, d\mu(t).$$ The novelty of our definition is that it contains the factor |t−x| q , which is of particular interest. Its properties are discussed and estimates are given. In particular, upper and lower bounds for generalized Christoffel functions with respect to generalized Jacobi weights are also provided.