Let x1,...,xn be real numbers, P(x)=pn(x−x1)...(x−xn), and Q(x)
be a polynomial of degree less than or equal to n. Denote by Δ(Q) the
matrix of generalized divided differences of Q(x) with nodes x1,...,xn
and by B(P,Q) the Bezout matrix (Bezoutiant) of P and Q. A relationship
between the corresponding principal minors, counted from the right-hand lower
corner, of the matrices B(P,Q) and Δ(Q) is established. It implies
that if the principal minors of the matrix of divided differences of a function
g(x) are positive or have alternating signs then the roots of the Newton's
interpolation polynomial of g are real and separated by the nodes of
interpolation.Comment: 15 page