Orthogonality of Quasi-Orthogonal Polynomials
Abstract
Given a sequence $\left \{ P_{n}\right \}_{n\geq0}$ of monic orthogonal
polynomials and a fixed integer $k$, we establish necessary and sufficient conditions so that the polynomials
$Q_{n}$, defined by
\begin{equation*}
Q_{n}(x) =P_{n}(x) + \sum \limits_{i=1}^{k-1} b_{i,n}P_{n-i}(x), \ \ n\geq 0,
\end{equation*}%
with $b_{i,n} \in \mathbb{C}$, and $b_{k-1,n}\neq 0$ for $n\geq k-1$,
also constitute a sequence of orthogonal polynomials. Therefore we solve the inverse problem
for linearly related orthogonal polynomials. The characterization turns out to be equivalent
to a recurrence formula for the coefficients $b_{i,n}$. The methods are based on
the relation between the corresponding Jacobi matrices as well as on
the interplay between Sturm's theorem on location of zeros of algebraic polynomials and
Favard's theorem on polynomial sequences satisfying three-term recurrence relations.
Various particular cases and the location of the zeros of $P_n$ and $Q_n$ are discussed.
polynomials and a fixed integer $k$, we establish necessary and sufficient conditions so that the polynomials
$Q_{n}$, defined by
\begin{equation*}
Q_{n}(x) =P_{n}(x) + \sum \limits_{i=1}^{k-1} b_{i,n}P_{n-i}(x), \ \ n\geq 0,
\end{equation*}%
with $b_{i,n} \in \mathbb{C}$, and $b_{k-1,n}\neq 0$ for $n\geq k-1$,
also constitute a sequence of orthogonal polynomials. Therefore we solve the inverse problem
for linearly related orthogonal polynomials. The characterization turns out to be equivalent
to a recurrence formula for the coefficients $b_{i,n}$. The methods are based on
the relation between the corresponding Jacobi matrices as well as on
the interplay between Sturm's theorem on location of zeros of algebraic polynomials and
Favard's theorem on polynomial sequences satisfying three-term recurrence relations.
Various particular cases and the location of the zeros of $P_n$ and $Q_n$ are discussed.
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