On Exceptional Sets of Asymptotic Relations for General Orthogonal Polynomials

Ambroladze, Amiran

doi:10.1006/jath.1995.1077

Cited by 22 publications

(30 citation statements)

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“…As a counterpart of Proposition 3.2(a) and (33), we may prove uniform convergence of quite a dense subsequence in some neighborhood of any`# 0 with geometric rate. This generalizes a result of Ambroladze [1,Corollaries 3 and 4], who studied the special case of real recurrence coefficients. |4 } a n } r n (z) } r n+1 (z)| 1Â(2n) , n 0, z # U.…”

Section: Local Uniform Convergence Of Bounded J-fractionssupporting

confidence: 70%

“…Let 4/N be infinite, = n # [0, 1], and suppose that the sequence of measure (+ n+= n ) n # 4 and (+ n+1&= n ) n # 4 converge weakly to + (0) , and to + (1) , respectively. Then the potentials of both limit measures coincide in 0 0 .…”

Section: N Th-root Asymptotic Behaviormentioning

confidence: 99%

“…We recall from [24,Chap. VIII] that the n th convergent of the continued fraction (1) ? n (z) | (2) may be rewritten as ?…”

Section: Introductionmentioning

confidence: 99%

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On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator

Beckermann¹

1999

Journal of Approximation Theory

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We study connections between continued fractions of type J and spectral properties of second order difference operators with complex coefficients. It is known that the convergents of a bounded J-fraction are diagonal Pade approximants of the Weyl function of the corresponding difference operator and that a bounded J-fraction converges uniformly to the Weyl function in some neighborhood of infinity. In this paper we establish convergence in capacity in the unbounded connected component of the resolvent set of the difference operator and specify the rate of convergence. Furthermore, we show that the absence of poles of Pade approximants in some subdomain implies already local uniform convergence. This enables us to verify the Baker Gammel Wills conjecture for a subclass of Weyl functions. For establishing these convergence results, we study the ratio and the n th root asymptotic behavior of Pade denominators of bounded J-fractions and give relations with the Green function of the unbounded connected component of the resolvent set. In addition, we show that the number of``spurious'' Pade poles in this set may be bounded. Academic Press

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Section: Local Uniform Convergence Of Bounded J-fractionssupporting

confidence: 70%

Section: N Th-root Asymptotic Behaviormentioning

confidence: 99%

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On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator

Beckermann¹

1999

Journal of Approximation Theory

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“…There is a version of (15) (see [1], Theorem 1$) giving uniform convergence in a neighbourhood of an arbitrary point of 0. That gives a corresponding version of (7) with uniform convergence.…”

Section: Proofsmentioning

confidence: 98%

Convergence Rates of Padé and Padé-Type Approximants

Ambroladze

Wallin

1996

Journal of Approximation Theory

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“…(1) The assumption S = Clos(Int(S))) for S = supp(µ) is strongly used, which does not allow outliers. (2) The assumption S = Clos(Int(S))) is strongly used also to give upper bounds for K µ ntot(m) (z, z) in supp(µ) at some distance from the boundary. This condition could be replaced by stepping to the relative interior of supp(µ) in S(µ) ∩ R d (in the real setting) or S(µ) in the complex setting.…”

Section: Approximation Of Christoffel-darboux Kernelsmentioning

confidence: 99%

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

et al. 2020

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Two central objects in constructive approximation, the Christoffel-Darboux kernel and the Christoffel function, are encoding ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel-Darboux kernel in C d , with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri)potential theory are widely used in the derivation of our perturbation formulas.

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On Exceptional Sets of Asymptotic Relations for General Orthogonal Polynomials

Cited by 22 publications

References 0 publications

On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator

On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator

Convergence Rates of Padé and Padé-Type Approximants

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

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