On Markov’s theorem on zeros of orthogonal polynomials revisited
Introduction
Markov’s theorem, dating back to the late 19th century, furnishes a method for obtaining information about zeros of orthogonal polynomials from the weight function related to orthogonality. Formally, adopting modern terminology, his result is stated as follows (see [22]):
Theorem 1.1 [22] Let {pn(x, t)} be a sequence of polynomials which are orthogonal on the interval with respect to the weight function ω(x, t) that depends on a parameter t, i.e.,
Suppose that ω(x, t) is positive and has a continuous first derivative with respect to t for x ∈ A, t ∈ B. Furthermore, assume that
converge uniformly for t in every compact subinterval of B. Then the zeros of pn(x, t) are increasing (decreasing) functions of t, t ∈ B, provided that
is an increasing (decreasing) function of x, x ∈ A.
Markov’s proof is based on the orthogonality relation (cf. [22, Eq. (2)]) together with the chain rule (cf. [22, Eq. (5)]), supposing that the zeros are defined implicitly as differentiable functions of the parameter. In addition, as an application of this result, Markov established that the zeros of Jacobi polynomials, which are orthogonal in with respect to the weight function are decreasing functions of α and increasing functions of β. Later, in 1939, Szegő, in his classical book [26, Theorem 6.12.1, p. 115], provided a different proof of Markov’s theorem. Szegő referred his proof of Theorem 1.1 in the following way [26, Footnote 31, p. 116]: “This proof does not differ essentially from the original one due to A. Markov, although the present arrangement is somewhat clearer.” Szegő’s reasoning (argument, approach) is based on Gauss mechanical quadrature, which was an approach that Stieltjes suggested to handle the problem, see [25, Section 5, p. 391]. In 1971, Freud (see [9, Problem 16, p. 133]) formulated a version of Markov’s theorem that is a little more general, considering sequences of polynomials orthogonal with respect to measures in the form . A proof of a result appears in Ismail [12, Theorem 3.2, p. 183 ] (see also in Ismail’s book [13, Theorem 7.1.1, p. 204]). Ismail’s argument of the proof is also based on Gauss mechanical quadrature. As consequence, Ismail provides monotonicity properties for the zeros of Hahn and Meixner polynomials (see [13, Theorem 7.1.2, p. 205]). Kroó and Peherstorfer [18, Theorem 1], in a more general context of approximation theory, extended Markov’s result to zeros of polynomials which have the minimal Lp-norm. Their approach is based on the implicit function theorem.
The main concern of this work derives from Markov’s classic 1886 theorem. This allows the approach to be tailored towards measures with continuous and discrete parts, thus extending Markov’s result. This point at issue was posed by Ismail in his book as an open problem [13, Problem 24.9.1, p. 660] (see also [12, Problem 1, p. 187]). The question is stated as follows:
Problem 1.1 Let μ be a positive and nontrivial Borel measure on a compact set . Assume that dμ(x, t) has the form
where dα(x, t) ≔ ω(x, t)dν(x) and ,1with t ∈ B, B being an open interval on . Determine sufficient conditions in order for the zeros of the polynomial Pn(x, t) to be strictly increasing (decreasing) functions of t.
The manuscript is organized in the following way: in Section 2 the main result is stated and proved; in Section 3 some conclusions are drawn from the main result, including Markov’s classic theorem, among others; finally, in Section 4, illustrative examples are given: in Sections 4.1 and 4.2 monotonicity properties of zeros of polynomials orthogonal with respect measures with discrete parts are investigated; in Section 4.3, sharp monotonicity properties involving the zeros of Gegenbauer–Hermite, Jacobi–Laguerre and Laguerre–Hermite orthogonal polynomials are derived.
Section snippets
Main results
The next result extends Markov’s theorem to measures with continuous and discrete parts, giving an answer to Problem 1.1. For a result in the context of polynomials which have minimal Lp-norm see [2, Theorem 1.1].
Theorem 2.1 Assume the notation and conditions of Problem 1.1. Assume further the existence and continuity for each x ∈ A and t ∈ B of (∂ω/∂t)(x, t) and, in addition, suppose that
converges at and
Markov’s theorem and its descendants
One can easily see that Markov’s classic theorem [22] (see also [26, Theorem 6.12.1, p. 115] and [13, Theorem 7.1.1, p. 204]) is derived from Theorem 2.1.
In Markov’s theorem, one can consider the end points of the interval of the orthogonality as functions of the parameter, i.e., and . From this, the following result can be stated:
Corollary 3.1 Assume the notation and conditions of Theorem 2.1 under the constraint dμ(x, t) . Furthermore, suppose that and are functions of
Sharp monotonicity properties of the zeros of orthogonal polynomials derived from Corollary 3.2
Suppose that where and with and . Let {pn} be the sequence of orthogonal polynomials with respect to dμ, i.e., Then, the zeros of the polynomial pn located on the left-hand side of are increasing functions of t, while the zeros of pn on the right-hand side of are decreasing functions of t, in view of Corollary 3.2.
Table 1 shows the monotonicity
Acknowledgments
The authors thank the valuable comments from the anonymous referees. Castillo is supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/101139/2014 and partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. Rafaeli is supported by the
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