Abstract
In this paper, we deal with a problem of positivity of linear functionals in the linear space ℙ of polynomials in one variable with complex coefficients. Some new results of connection relations between the corresponding sequences of monic orthogonal polynomials of classical character are established.
Highlights
E linear functional U is positive definite if and only if Δn(U) > 0, for every integer n ≥ 0, where Δn(U) is the Hankel determinants of order n of U
Using Oppenheim’s inequality we prove that, for any pair (μ, c) ∈ ]0, +∞[ × R and any positive-definite linear functional U, the linear functional U(μ, c) given by (μ − 1)U(μ, c) − (x − c)(U(μ, c))′ μU is positive definite
A linear functional U is said to be quasi-definite if we can associate with it a monic polynomial sequence (MPS) Bnn ≥ 0 with degBn n, n ≥ 0, such that
Summary
E linear functional U is positive definite if and only if Δn(U) > 0, for every integer n ≥ 0, where Δn(U) is the Hankel determinants of order n of U (see [5]). 2. Orthogonality and Positive-Definite Linear Functional Character A linear functional U is said to be quasi-definite (regular) if we can associate with it a monic polynomial sequence (MPS) Bnn ≥ 0 with degBn n, n ≥ 0, such that
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