Bounds on orthogonal polynomials and separation of their zeros

Abstract

Let {pn} denote the orthonormal polynomials associated with a measure μ with compact support on the real line. Let μ be regular in the sense of Stahl, Totik, and Ullmann, and I be a subinterval of the support in which μ is absolutely continuous, while μ′ is positive and continuous there. We show that boundedness of the {pn} in that subinterval is closely related to the spacing of zeros of pn and pn−1 in that interval. One ingredient is proving that "local limits" imply universality limits. Abstract. Research supported by NSF grant DMS1800251 Research supported by NSF grant DMS1800251 1. Results Let μ be a finite positive Borel measure with compact support, which we denote by supp[μ]. Then we may define orthonormal polynomials pn (x) = γnx n + ..., γn > 0, n = 0, 1, 2, ... satisfying the orthonormality conditions ∫ pnpmdμ = δmn. The zeros of pn are real and simple. We list them in decreasing order: x1n > x2n > ... > xn−1,n > xnn. They interlace the zeros yjn of pn : pn (yjn) = 0 and yjn ∈ (xj+1,n, xjn) , 1 ≤ j ≤ n− 1. It is a classic result that the zeros of pn and pn−1 also interlace. The three term recurrence relation has the form (x− bn) pn (x) = an+1pn+1 (x) + anpn−1 (x) , where for n ≥ 1, an = γn−1 γn = ∫ xpn−1 (x) pn (x) dμ (x) ; bn = ∫ xpn (x) dμ (x) . 1 2 ELI LEVIN AND D. S. LUBINSKY Uniform boundedness of orthonormal polynomials is a long studied topic. For example, given an interval I, one asks whether sup n≥1 ‖pn‖L∞(I) <∞. There is an extensive literature on this fundamental question see for example [1], [2], [3], [4], [12]. In this paper, we establish a connection to the distance between zeros of pn and pn−1. The results require more terminology: we let dist (a,Z) denote the distance from a real number a to the integers. We say that μ is regular (in the sense of Stahl, Totik, and Ullmann) if for every sequence of non-zero polynomials {Pn} with degree Pn at most n, lim sup n→∞ ( |Pn (x)| (∫ |Pn| dμ )1/2 )1/n ≤ 1 for quasi-every x ∈supp[μ] (that is except in a set of logarithmic capacity 0). If the support consists of finitely many intervals, and μ′ > 0 a.e. in each subinterval, then μ is regular, though much less is required [15]. An equivalent formulation involves the leading coeffi cients {γn} of the orthonormal polynomials for μ : lim n→∞ γ n = 1 cap (supp [μ]) , where cap denotes logarithmic capacity. Recall that the equilibrium measure for the compact set supp[μ] is the probability measure that minimizes the energy integral ∫ ∫ log 1 |x− y| (x) dν (y) amongst all probability measures ν supported on supp[μ]. If I is an interval contained in supp[μ], then the equilibrium measure is absolutely continuous in I, and moreover its density, which we denote throughout by ω, is positive and continuous in the interior I of I [13, p.216, Thm. IV.2.5]. Given sequences {xn} , {yn} of non-0 real numbers, we write xn ∼ yn if there exists C > 1 such that for n ≥ 1, C−1 ≤ xn/yn < C. Similar notation is used for functions and sequences of functions. Our main result is BOUNDS ON ORTHOGONAL POLYNOMIALS, DEC 31, 2020 3 Theorem 1.1 Let μ be a regular measure on R with compact support. Let I be a closed subinterval of the support and assume that in some open interval containing I, μ is absolutely continuous, while μ′ is positive and continuous. Let ω be the density of the equilibrium measure for the support of μ. Let A > 0. The following are equivalent: (a) There exists C > 0 such that for n ≥ 1 and xjn ∈ I, (1.1) dist (nω (xjn) (xjn − xj,n−1) ,Z) ≥ C. (b) There exists C > 0 such that for n ≥ 1 and yjn ∈ I, (1.2) dist (nω (yjn) (yjn − yj,n−1) ,Z) ≥ C. (c) Uniformly for n ≥ 1 and x ∈ I, (1.3) ‖pn−1‖L∞[x−An ,x+An ] ‖pn‖L∞[x−An ,x+An ] ∼ 1. (d) There exists C > 0 such that for n ≥ 1 and x ∈ I, (1.4) ‖pn−1‖L∞[x−An ,x+An ] ‖pn‖L∞[x−An ,x+An ] ≤ C. Moreover, under any of (a), (b), (c), (d), we have (1.5) sup n≥1 sup x∈I ∣∣∣|x− bn| pn (x)∣∣∣ <∞. Remarks (a) The main idea behind the proof is that universality limits and "local" limits give |pn−1 (yj,n−1) pn (yjn)| |sin [πnω (yjn) (yjn − yj,n−1)] + o (1)| ∼ 1, uniformly in j, n, while pn has a local extremum at yjn. (b) We could replace xj,n−1 − xjn in (1.1) by xj,n−1 − xj,n+k, for any fixed integer k (see Lemma 4.1). (b) Under additional assumptions, involving the spacing of zeros of pn and pn−2, we can remove the factor |x− bn| in (1.5): Theorem 1.2 Let μ be a regular measure on R with compact support. Let I be a closed subinterval of the support and assume that in some open interval containing I, μ is absolutely continuous, while μ′ is positive and continuous. Let ω be the density of the equilibrium measure for the support of μ. Let A > 0. Assume that (1.1) holds in I. The following are equivalent: (a) There exist C1 > 0 such that for n ≥ 1 and xjn ∈ I, (1.6) |n (xjn − xj−1,n−2)| ≥ C1 |xjn − bn−1| . 4 ELI LEVIN AND D. S. LUBINSKY (b) Uniformly for x ∈ I and n ≥ 1, (1.7) ‖pn‖L∞[x−An ,x+An ] ∼ 1. (c) (1.8) sup n≥1 ‖pn‖L∞(I) <∞. Remark We note that because of the interlacing, both xjn and xj−1,n−2 belong to the interval (xj,n−1, xj−1,n−1). Two important ingredients in our proofs are universality and local limits. The so-called universality limit involves the reproducing kernel