Sobolev orthogonal polynomials with a small number of real zeros
The first part of the paper deals with the number of real zeros of polynomials \(P^ \lambda_ n\), \(n= 0,1,2,\dots\), which are orthogonal with respect to the Sobolev inner product \[ \langle f,g\rangle:= \int^ 3_{-1} f(x) g(x) dx+ \lambda \int^ 1_{-1} f'(x) g'(x) dx+ \int^ 3_ 1 f'(x) g'(x) dx, \] where \(\lambda\) denotes a fixed nonnegative real number. It is proved: If \(n\) is odd and \(\lambda\) sufficiently large, then \(P^ \lambda_ n\) has exactly one real zero: it is in (1,3). If \(n\) is even, \(n\geq 2\), and \(\lambda\) sufficiently large, then \(P^ \lambda_ n\) has exactly two real zeros: one in (1,3), and the other one in \((-3,-1)\). In the second part of the paper this result is generalized with respect to the Sobolev inner product \[ \langle f,g\rangle:= \int^ a_{-1} f(x) g(x) d\psi_ 0(x)+ \lambda \int^ 1_{-1} f'(x) g'(x) d\psi_ 1(x)+ \int^ a_ 1 f'(x) g'(x) d\psi_ 1(x), \] where \(1< a\leq +\infty\), \(\lambda\geq 0\) and \(\psi_ 0\), \(\psi_ 1\) are suitable distribution functions on \([-1,a]\).
scientific article; zbMATH DE number 636072