Biorthogonal polynomials and zero-mapping transformations

• Published in: Computers & mathematics with applications (1987) Vol. 33; no. 1; pp. 129 - 143
• Main Authors: Iserles, A., Nørsett, S.P.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 1997
• Subjects: Biorthogonal polynomials; Special functions; Strict sign consistency; Strict sign consistency; Biorthogonal polynomials; Special functions
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/S0898-1221(96)00225-8

The authors have presented in [1] a technique to generate transformations T of the set P n of n th degree polynomials to itself such that if if p ϵ P n has all its zeros in ( c, d) then T{ p} has all its zeros in ( a, b), where ( a, b) and ( c, d) are given real intervals. The technique rests upon the derivation of an explicit form of biorthogonal polynomials whose Borel measure is strictly sign consistent and such that the ratio of consecutive generalized moments is a rational [ 1 1 ] function of the parameter. Specific instances of strictly sign consistent measures that have been debated in [1] include x μ d ψ( x), μ x d ψ( x) and x log q μ d ψ( x), q ϵ (0, 1). In this paper, we identify all measures ψ such that their consecutive generalized moments have a rational [ 1 1 ] quotient, thereby characterizing all possible zero-mapping transformations of this kind.