On the asymptotic behavior of Jacobi polynomials with first varying parameter

Szehr, Oleg; Zarouf, Rachid

doi:10.1016/j.jat.2022.105702

Cited by 4 publications

(3 citation statements)

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“…The slowest decay of B(k) occurs at the boundary k = α −1 0 n. Here the supremum | B(k)| is attained and corresponds to the l A ∞ -norm. In this situation we can recover some of the findings of [SZ,Proposition 4]. We find that the boundary behavior as n gets large (at…”

Section: Lower Estimatessupporting

confidence: 83%

“…The most complicated case turns out the boundary case p = 4, which is treated separately in the end. We notice that the upper bound in Theorem 1 is sharp for p = ∞ as a consequence of [SZ,Theorem 2,point (2)]. In that reference the behavior of the largest coefficient, sup k≥0 | B(k)| is analyzed and it is shown that to first order we have sup k≥0 B(k) ∼ n −1/3 .…”

Section: Lower Estimatesmentioning

confidence: 96%

“…(1) This is a direct application of [SZ,Theorem 2,point (3)]. We recapitulate the main steps for completeness.…”

Section: Upper Estimatesmentioning

confidence: 97%

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lp-norms of Fourier coefficients of powers of a Blaschke factor

Szehr

Zarouf

2020

JAMA

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We determine the asymptotic behavior of the l p -norms of the sequence of Taylor coefficients of b n , where b = z−λ 1−λz is an automorphism of the unit disk, p ∈ [1, ∞], and n is large. It is known that in the parameter range p ∈ [1, 2] a sharp upper boundholds. In this article we find that this estimate is valid even when p ∈ [1, 4). We prove thatWe prove that our upper bounds are sharp as n tends to ∞ i.e. they have the correct asymptotic n dependence.the elementary Blaschke factor corresponding to λ. Clearly |b λ (z)| = 1 is equivalent to z ∈ ∂D. For any n we have that B = b n is a bounded, holomorphic on D and as such posses a natural identification with its boundary behavior on ∂D [NN]. It is well known that the Taylor-and Fourier-coefficients of such functions can be identified [NN] and we will use these terms interchangeably in what follows. Let B = k≥0 B(k)z k denote the Taylor expansion of B = b n . We writefor the usual l p -norm of the sequence of Taylor coefficients of B. In the limit of large p we set ||B|| l∞ := sup k | B(k)|. We observe that our l p -norms only depend on the absolute

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Section: Lower Estimatessupporting

confidence: 83%

Section: Lower Estimatesmentioning

confidence: 96%