Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle

Mate, Attila; Nevai, Paul; Тотик, Вилмос

doi:10.1007/bf01890022

Cited by 92 publications

(32 citation statements)

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“…For the significance of Nevai's class M and references to the numerous investigations thereof, see [4], [7], [9], and [10].…”

Section: Introductionmentioning

confidence: 99%

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Singularly continuous measures in Nevai’s class 𝑀

Lubinsky¹

1991

Proc. Amer. Math. Soc.

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Abstract.Let dv be a nonnegative Borel measure on \-n, n], with 0 < f*n du < co and with support of Lebesgue measure zero. We show that there exist {Vj}%¡ C (0, oo) and {tj}jZ\ C (-it, n) such that if 00 dß(6) :=Y/1jdu(d + tj), 6 e[-n , ti], j=i (with the usual periodic extension dv(d ± 2n) = dv(6)), then the leading coefficients {Kn(dß)}^L0 of the orthonormal polynomials for dß satisfy \im>K"(dp)/Kn+x(dp)= 1.As a consequence, we obtain pure singularly continuous measures da on [-1,1] lying in Nevai's class M .

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“…For the significance of Nevai's class M and references to the numerous investigations thereof, see [4], [7], [9], and [10].…”

Section: Introductionmentioning

confidence: 99%

“…(e) For results on the distribution of zeros of orthogonal polynomials on the unit circle and [-1, 1], see [3,7,8,10,11,13,16,17].…”

Section: Introductionmentioning

confidence: 99%

Singularly continuous measures in Nevai’s class 𝑀

Lubinsky¹

1991

Proc. Amer. Math. Soc.

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“…where C is an absolute constant independent of n and m. For the proof of (4)- (6) see Chapter 1 of [7], for that of (7) see Theorem 2 in [13]. Combining (4)- (7) and using Theorem 2, we get…”

Section: Introductionmentioning

confidence: 88%

“…From this, it follows that for each trigonometric polynomial T m of degree ≤ m, we have 1 2π (13) and (iii) of admissibility, it follows that…”

Section: Proof Of Theoremmentioning

confidence: 95%

Weak Convergence of Varying Measures and Hermite—Padé Orthogonal Polynomials

Ysern

Lagomasino

1999

Constr. Approx.

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It is known that the common denominator of the Hermite-Padé approximants of a mixed Angelesco-Nikishin system shares orthogonality relations with respect to each function in the system. It is less known that they also satisfy full orthogonality with respect to a varying measure. This problem motivates our interest in extending the class of varying measures with respect to which weak asymptotics of orthogonal polynomials takes place. In particular, for the case of a Nikishin system, we prove weak asymptotics of the corresponding varying measures.

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“…Rakhmanov [14,15] (see also [13]) proved a deep theorem about orthogonal polynomials on the unit circle that translates to Here is how Denisov [6] used this to prove Denisov-Rakhmanov Theorem. If dµ = f (x) dx + dµ 0 , f (x) > 0 a.e.…”

Section: Some Applicationsmentioning

confidence: 99%

Sturm Oscillation and Comparison Theorems

Simon

Sturm-Liouville Theory

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Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle

Cited by 92 publications

References 7 publications

Singularly continuous measures in Nevai’s class 𝑀

Singularly continuous measures in Nevai’s class 𝑀

Weak Convergence of Varying Measures and Hermite—Padé Orthogonal Polynomials

Sturm Oscillation and Comparison Theorems

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