On Christoffel type functions for \(L_m\) extremal polynomials. I (Q1976274)

Let \(m\in \mathbb{N}\) be even and \(M:=\{ 0,2,4,\dots{},m-2\} .\) For \(n\in \mathbb{N}\) let \(\mathbf{P}_{n}\) be the set of all polynomials of degree \(\leq n\) and \(\mathbf{P}_{n}^{\ast }\) the subset of \(\mathbf{P}_{n}\) formed by polynomials having only real zeros. Let \(x\in \mathbb{R}\) and \(\mathbf{P}_{n}^{\ast }( x) :=\{ P\in \mathbf{P}_{n}^{\ast }:P( x) =1\} .\) For \(j\in M\) denote \[ A_{j}( P,x;t) :=A_{jnm}( P,x;t) :=\frac{1}{j!}( t-x) ^{j}B_{j}( P,x;t) ( P( t)) ^{m}, \] where \(P\in \mathbf{P}_{n-1},\) \(P( x) =1,\) \(B_{j}( P,x;\cdot) \in \mathbf{P}_{m-j-2}\) and \(A_{j}^{( i) }( P,x;x) =\delta _{ij},\) \(i=0,1,2,\dots{},m-2.\) The function defined by \(\lambda _{jnm}( d\alpha ,x) =\inf_{P\in \mathbf{P}_{n-1}^{\ast }( x) }\int_{\mathbb{R} }A_{j}( P;x;t) d\alpha ( t) ,\) \(x\in \mathbb{R},\) where \(\alpha :\mathbb{R}\rightarrow \mathbb{R}\) is a nondecreasing function with infinitely many points of increase such that all moments of \(d\alpha\) are finite is called Christoffel type function with respect to \(d\alpha \). The author proves that the preceeding infinum is attained for a unique polynomial \(P\in \mathbf{P}_{n-1}^{\ast }( x) \) with effective degree \(\geq n-2,\) and presents some properties of the functions \(\lambda_{jnm}( d\alpha ,\cdot) ,\) \(j\in M.\)